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On $[0,1]$, for $n\ge 2$, these continuous functions have support $[0,2/n]$, height $n$, and integral one. Their pointwise limit is zero: at zero every value vanishes, while every fixed positive argument eventually leaves the support. Thus pointwise convergence of continuous functions on a compact interval does not suffice for convergence of their integrals. The mass becomes more concentrated as its height grows.

For arbitrary topological spaces $X_i$, an index set $I$, a sequence $x^{(n)}\in {\prod }_i{X}_i$, and a point $x$ in that Cartesian product, convergence in a product space to $x$ in the product topology is equivalent to coordinatewise convergence in every factor. No countability or Hausdorff assumption on the factors is needed. For a product space of copies of the real line, regarding each point as a function $I\to \mathbb{R}$ identifies this with pointwise convergence.

Proof outline: the forward implication follows because each projection map is continuous. Conversely, given a basic neighborhood $B(F,U)$ of $x$, each of its finitely many coordinate tests holds for all $n\ge N_i$. Taking $N={\max }_{i\in F}{N}_i$ makes all tests hold simultaneously; if $F$ is empty, any threshold works. This proves eventual membership in every neighborhood, rather than only in a sampled collection of cylinders. The same equivalence holds for nets: a directed set supplies a common upper bound for finitely many thresholds. Sequences alone need not characterize the whole topology of an arbitrary product space.

A Fourier series describes a periodic function through sines and cosines whose frequencies are integer multiples of one base frequency. These Fourier harmonics give a useful alternative to describing the value at every input: each Fourier coefficient measures how much of one oscillating pattern is present. For a real, integrable function of period $2\pi$, the real form isThe cosine formula includes $n=0$; the sine formula uses $n\ge 1$. Thus $a_0/2$ is the mean value over a period, not $a_0$ itself. The symbol $\sim$ associates a function with its Fourier coefficients; an equality requires a specified kind of convergence, such as pointwise convergence. For general period $T>0$, replace $nx$ by $2\pi nt/T$ and the coefficient prefactor by $2/T$. Fourier orthogonality explains why integrating against one mode removes the others. ^[ref].

For a finite example, takeIts mean is $1/2$, its real Fourier coefficients are $a_0=1$, $a_1=1$, $b_2=1/2$, and all other $a_n,b_n$ vanish. At $x=\pi /4$, the three contributions add to $1/2+1/\sqrt{2}+1/2=1+1/\sqrt{2}\approx 1.70710678$. The function is already a trigonometric polynomial, so retaining these modes reconstructs it exactly. Here $x$ is an angular coordinate in radians; if it instead represents time, a base angular frequency must supply the corresponding units.

The change of viewpoint is useful because many linear problems act separately on these modes. The second derivative maps $\sin (nx)$ to $-n^2\sin (nx)$ and maps $\cos (nx)$ in the same way. Consequently a differential equation that is complicated for a whole profile can become a simpler calculation for each Fourier harmonic. A linear time-invariant system similarly has a separate response to each complex exponential. Superposition combines the responses when the relevant sums and operations are justified. ^[ref].

The infinite case contains a subtlety absent from the finite example. Under pointwise convergence of a piecewise smooth Fourier series, the value at a jump discontinuity is the average of the two one-sided limits. For a square-integrable function, the complex Fourier series instead converges in the $L^2$ norm: the integrated squared error tends to zero. Neither statement says that the error becomes small everywhere at once. A square wave shows the difference vividly through the Gibbs phenomenon.

A historical source for this connection between profiles and modes is Joseph Fourier's 1822 book Théorie analytique de la chaleur. Chapter III develops expansions in sines and cosines while studying the propagation of heat. The heat equation gives a continuing application of this approach. ^[ref].

For the degree-$N$ partial sum $P_N(z)=1+z+\cdots +z^N$ of the geometric series, the finite identity $(1-z)P_N(z)=1-z^{N+1}$ givesFor any fixed $0<r<1$, the triangle inequality then gives the uniform error boundEquality is attained at the positive real point $z=r$, so this is the exact worst-case error on that closed disk. It also proves uniform convergence on each such disk, not just pointwise convergence.

Given a positive tolerance $\epsilon$, choosing $N$ with $r^{N+1}/(1-r)\le \epsilon$ turns a convergence statement into a finite stopping rule. For $\epsilon =10^{-3}$, degree ten suffices throughout $|z|\le 1/2$. Nearer the boundary, $r=0.9$ requires degree $87$ and $r=0.99$ requires degree $1145$. The required degree depends on both the tolerance and the region, not just on the name of the function.

The Gibbs phenomenon is the persistent overshoot near a jump discontinuity in ordinary truncated Fourier series. For this square wave, retain $M$ nonzero odd modes:As $M$ grows, the first peak approaches the jump while its height tends to approximately $1.17897974$. This is an excess of about $0.17898$ above the upper level $1$, or $8.949\%$ of the jump size $2$. The narrower region does not imply a vanishing maximum error. At the jump itself every $Q_M(0)=0$. These observations are compatible with pointwise convergence and $L^2$ convergence but exclude uniform convergence to the discontinuous square wave. ^[ref].

For this particular sum, its derivative is $Q_M^{\prime }(x)=(2/\pi )\sin (2Mx)/\sin x$ away from multiples of $\pi$, so the first positive peak is exactly at $x=\pi /(2M)$. There,This is a midpoint Riemann sum tending to $2\mathrm{Si}(\pi )/\pi$, where $\mathrm{Si}$ is the sine integral. It gives a short explanation of the nonzero limiting excess.

The phenomenon matters whenever a sharp boundary is represented by a limited set of Fourier harmonics: a visible ripple can be a property of the approximation rather than an oscillation in the original profile. Josiah Willard Gibbs reported the phenomenon in his 1899 letter “Fourier's Series” in Nature, volume 59, page 606. This is a documented historical report, without a claim of first discovery. ^[ref].

A sequence in L2 space converges in its L2 norm when $\int |f_n-f{|}^{2}d\mu \to 0$. This concerns the integral of the squared error and need not imply pointwise convergence. In particular a complete Fourier sine basis reconstructs each odd L2 function in this norm, while pointwise claims about its Fourier series require additional hypotheses.

Let $z_0$ be a boundary point of a compact convex set and choose exterior points ${\zeta }_n\to z_0$. Approximate $(z-{\zeta }_n{)}^{-1}$ uniformly on the compact set by a polynomial with error at most $1/n$. At each fixed $z\ne z_0$, the reciprocal converges and so does the approximating polynomial. On a boundary accumulating at $z_0$, uniform convergence to the singular reciprocal is impossible because each polynomial is bounded there. This distinguishes pointwise convergence from uniform convergence without confusing the two norms.

Augustin-Louis Cauchy's importance lies in the interaction of foundations, methods and applications. He made the justification of limits, series and integrals a programme for teaching and research; developed complex analysis into a source of powerful general methods; and contributed to permutation groups, geometry and continuum mechanics. His work is best assessed through particular achievements and their limitations, rather than by treating every theorem now bearing his name as a modern statement he already possessed. His engineering training and subsequent mathematical career also make a rigid separation of pure and applied work historically unhelpful.^[ref]

The Cours d'analyse by Augustin-Louis Cauchy (1821) makes convergence a condition requiring argument. A series is studied through its partial sums; a limit describes a variable approaching a fixed value; an infinitesimal is a variable tending to zero, rather than simply a mysterious fixed quantity smaller than every positive number. The introduction explicitly challenges unrestricted transfers of rules from finite expressions to infinite ones. This is not an abandonment of calculation: it seeks to identify when calculations are valid.^[ref]

For example, the Cauchy criterion for series asks whether tails can all be made small:This is a modern formulation of the convergence requirement, with the completeness of the real numbers understood. Merely requiring $a_n\to 0$ is weaker. In the harmonic series, grouping the terms from $2^j+1$ to $2^{j+1}$ gives a block of at least $1/2$, so its tails cannot satisfy the criterion. In contrast, the root test compares a tail with a geometric series: if $\mathrm{limsup}|a_n{|}^{1/n}<q<1$, then eventually $|a_n|\le q^n$, which proves absolute convergence. Such arguments explain the usefulness of convergence tests instead of reducing Augustin-Louis Cauchy's contribution to a vocabulary of definitions.

His Résumé des leçons sur le calcul infinitésimal by Augustin-Louis Cauchy (1823) extends the programme to differentiation and integration. For a continuous function on a compact interval, an integral can be approached through sums of values times subinterval lengths. A modern justification uses uniform continuity: for a partition of mesh at most $\delta$, two choices of sample points give sums differing by at most $(b-a){\omega }_f(\delta )$, where the modulus of continuity ${\omega }_f(\delta )$ tends to zero. Refinement then establishes the common limit. This reconstruction makes clear why control of error, rather than a geometrical picture of area alone, is decisive. It does not attribute the later full Riemann integral theory for discontinuous functions to the 1823 text.^[ref]

The achievement has limits. The 1821 Sum theorem of Cauchy asserts continuity of sums of convergent series of continuous functions in terms that, under a pointwise reading, are insufficient. The original proof treats the tail as negligible without establishing the uniform control needed near the point concerned. A modern diagnostic is $f_n(x)=x^n$ on $[0,1]$: every $f_n$ is continuous, but its pointwise limit is zero for $x<1$ and one at $1$. Thus a proof must distinguish pointwise convergence from uniform convergence. Augustin-Louis Cauchy's use of variable and infinitesimal language deserves historical interpretation, but it does not license silently importing modern hypotheses into his printed theorem.^[ref]

In complex analysis, the Mémoire sur les intégrales définies, prises entre des limites imaginaires by Augustin-Louis Cauchy (1825) exemplifies a new productive setting for integration. A contour integral depends on a path in the complex plane; deforming that path and accounting for isolated singularities can evaluate otherwise difficult definite integrals. In modern form, Cauchy's integral theorem applies to a holomorphic function on a suitable domain and Cauchy integral formula states, for a circle enclosing $z$ and lying with its interior in that domain,The importance is structural: local values and derivatives are determined by data on a surrounding path. Historical proofs and hypotheses should be distinguished from later general forms of the theorem.^[ref]

A short modern consequence measures the power of this method. For an entire function bounded by $M$, the derivative version of Cauchy integral formula on a circle of radius $R$ gives $|f^{\prime }(z)|\le M/R$. Since $R$ is arbitrary, $f^{\prime }=0$, proving Liouville's theorem. If a nonconstant polynomial $P$ had no root, then $1/P$ would be an entire function, bounded outside a large disc by the leading term and inside the disc by compactness. Liouville's theorem would force it to be constant, a contradiction. This proves the fundamental theorem of algebra. The chain is an assessment of the reach of the methods, not a claim that Augustin-Louis Cauchy alone originated every named result.

Augustin-Louis Cauchy's algebra concerns concrete permutations as well as analytical formulas. His 1845 investigations developed the theory of substitutions; Cauchy theorem for groups says that a prime number dividing a finite group order occurs as an element order. One can appreciate its generality from a modern argument: rotate the coordinates of the tuples $(g_1,\dots ,g_p)$ satisfying $g_1\cdots g_p=e$. Rotation preserves this condition, since $g_2\cdots g_p{g}_1=g_1^{-1}(g_1\cdots g_p)g_1=e$. There are $|G{|}^{p-1}$ such tuples; nonfixed rotation orbits have $p$ elements, and fixed tuples correspond to $g^p=e$. When $p\mid |G|$, the number of fixed tuples is divisible by $p$, so besides $e$ there is an element of element order $p$. This is explanatory mathematics, not Augustin-Louis Cauchy's original proof. The dating itself needs care: Peter Michael Neumann's archival argument places the major burst of work in 1845 despite an 1844 date traditionally attached to a volume.^[ref]

Finally, Augustin-Louis Cauchy's work on elasticity shows the strength of making a local mathematical object out of physical interactions. Instead of describing only a force perpendicular to a surface, one studies the traction on a plane through a point, with its direction dependent on the plane's unit normal. In modern notation the relation is $\mathbf{t}(\mathbf{n})=\mathbf{\sigma }\mathbf{n}$, where the Cauchy stress tensor contains the necessary directional information. Together with balance of linear momentum, this yields equations for the internal motion of a body. The modern tensor notation is later explanatory language; the conceptual contribution is the reduction of the forces on arbitrary internal planes to a finite collection of local components.^[ref]

Augustin-Louis Cauchy's lasting contribution was a powerful research programme, not a finished foundation. His insistence on justified passage to a limit reshaped analysis, while complex integration, substitutions and local stresses opened methods whose value extended beyond their first applications. The inadequacy of a pointwise convergence hypothesis, the need to state domains in complex analysis, and the later construction of the real numbers show that nineteenth-century rigour remained a collective and revisable achievement. These qualifications strengthen an assessment of his influence by identifying what later mathematicians could extend and repair.

Standards of mathematical proof changed between the seventeenth and nineteenth centuries because both the objects studied and the operations performed on them changed. A finite geometrical construction, an infinite expansion, and an abstract group raise different questions about justification. There was no universal seventeenth-century indifference to rigour followed by its sudden nineteenth-century invention. The more defensible thesis is that successful methods repeatedly exposed unstated assumptions; later work made some of those assumptions into explicit definitions, hypotheses and existence arguments.

Pierre de Fermat's adequality illustrates the distinction between an effective method and its justification. For $F(a)=a(L-a)$, comparing neighbouring expressions givesDividing by $e$ and suppressing the remaining term in $e$ produces $a=L/2$. There is an independent finite verification:Thus the result can be secure even when the general procedure raises a question: why is the increment first nonzero for division and then discarded? In the parabola construction of question 2, the same issue appears in replacing a strict geometrical comparison by adequality and extracting the subtangent. A historical explanation must describe this operation on its own terms rather than silently equate it with a modern derivative.^[ref]

Isaac Newton and Gottfried Wilhelm Leibniz supplied different languages for the new calculus. Isaac Newton's Introduction to the Quadrature of Curves by Isaac Newton, first published with the 1704 Opticks, presents quantities generated by motion and considers their rates of change. Its mature language of first and last ratios seeks exact relations as increments vanish; it should not be confused with simply asserting that a small error is harmless. Gottfried Wilhelm Leibniz's 1684 Nova methodus pro maximis et minimis by Gottfried Wilhelm Leibniz presents a calculus of differentials and operational rules. Their common problem was to explain why procedures using vanishing increments establish exact results, while their chosen objects and modes of explanation differed.^[ref][ref]

George Berkeley's The Analyst by George Berkeley (1734) exposes the logical pressure point. For the cube,The finite identity does not become a legal division at $h=0$. The modern repair is to keep $h\ne 0$ throughout the identity and prove that its right-hand side tends to $3x^2$. That limit is different from a quotient evaluated at zero. George Berkeley's criticism therefore concerns the grounds of inference, not whether the established rules often produce correct answers. His theological polemic also reminds us that standards of mathematical proof could be contested outside specialist calculation.^[ref]

The infinite introduced another burden. Joseph Fourier's 1822 use of trigonometric series to represent arbitrary functions greatly extended what calculations in mathematical physics could express. But deriving Fourier coefficients by orthogonality only shows what those coefficients must be if the proposed operations are legitimate. It does not prove convergence, equality with the initial data, or interchange of an integral and an infinite sum. The example of a constant expanded in a Fourier sine series makes a hidden qualification visible: the terms all vanish at the endpoints, even when the assigned data do not. The correct theorem must specify the domain and mode of representation.^[ref]

Peter Gustav Lejeune Dirichlet's 1829 memoir supplies a particularly precise example of changing proof obligations. It criticises a convergence argument of Augustin-Louis Cauchy's which used complex substitution into an arbitrary function and an insufficient comparison between signed series. Peter Gustav Lejeune Dirichlet supplies the instructive pairThe first series converges by the alternating series test; the second diverges because their difference is the harmonic series. Nevertheless $b_n/a_n=1+(-1{)}^n/\sqrt{n}\to 1$. Thus asymptotic equality of terms does not transfer convergence between conditionally convergent signed series. A comparison principle valid for positive terms needs additional justification when cancellation is involved. The example refutes a mode of inference rather than merely correcting a numerical answer.^[ref]

Augustin-Louis Cauchy's own foundations programme is part of this change. The 1821 Cours d'analyse by Augustin-Louis Cauchy makes limits, continuity and series convergence central to elementary reasoning. Yet its Sum theorem of Cauchy also shows why announcing rigour does not make every proof adequate. In modern terminology, continuous functions can have a discontinuous pointwise limit. An explicit series version isEach term is a polynomial and the partial sum is $x-x^{N+1}$. The convergence does not hold in uniform convergence: with $x_N=1-1/N$, the error $x_N^{N+1}$ tends to $e^{-1}$ rather than zero. The modern uniform limit theorem repairs the inference by requiring one bound on the error to work throughout the domain. This example diagnoses the missing control under a pointwise reading; it does not settle every historical interpretation of Augustin-Louis Cauchy's infinitesimal language.^[ref]

The issue is the order of dependence in quantifiers. Pointwise convergence permits the threshold $N$ to depend on both the error tolerance and the point. Uniform convergence requires a threshold depending only on the tolerance. If $f_N$ is continuous near $x_0$ and the convergence is uniform, choose $N$ so that $|f-f_N|<\epsilon /3$ nearby, then use the continuity of this fixed $f_N$ to make $|f_N(x)-f_N(x_0)|<\epsilon /3$. The triangle inequality proves $|f(x)-f(x_0)|<\epsilon$. Writing out this proof reveals what a vague statement that the tail is small leaves unproved.

Another nineteenth-century development concerns the foundations of the real numbers themselves. Richard Dedekind's Continuity and Irrational Numbers by Richard Dedekind (1872) asks for an arithmetical account of the continuity often borrowed from a drawn line. A Dedekind cut separates the rational numbers into lower and upper classes. For instance,defines a gap in the rational numbers corresponding to $\sqrt{2}$. The cut is determined without assuming that a rational number fills it. The extended number system then supports existence theorems that a geometrical picture alone had suggested.^[ref]

This matters for apparently elementary proofs. Suppose $f$ is continuous on $[a,b]$ with $f(a)<0<f(b)$. Repeated bisection constructs nested intervals whose endpoint values have opposite signs and whose lengths tend to zero. To conclude that there is a common limiting point, one needs the completeness of the real numbers. Continuity then forces its value to be zero. Over the rational numbers, $f(q)=q^2-2$ changes sign between $1$ and $2$ but has no rational zero. Thus even a proof that avoids pictures must explain the existence of its limiting object. Richard Dedekind's account makes that requirement an explicit part of the mathematical setting, rather than an intuition about a continuous line.^[ref]

There is a parallel change in algebra. Heinrich Weber's 1882 definition in question 6 describes arbitrary elements through laws of composition. To assess such a definition one proves implications between axioms. The passage's finite set hypothesis implies that cancellative multiplication has an identity element and inverse elements; without that hypothesis, the positive integers under addition supply a counterexample. This is a new kind of proof obligation: one must establish that a proposed abstract definition has the intended consequences, rather than rely on their familiarity in particular numerical or permutation examples.^[ref]

The principal change was towards explicit control of operations, hypotheses and existence. Effective symbolic algorithms persisted, and geometric reasoning remained valuable. What became increasingly demanding was the passage from success in examples to a general claim: an infinitesimal step needed justification, an infinite expansion needed a convergence theorem, a limiting point needed a complete number system, and an abstract definition needed its consequences proved. The examples also prevent a simple progress story: nineteenth-century mathematicians disagreed about admissible reasoning, and a programme devoted to rigour could itself become the object of a successful critique.

False. A typewriter sequence sweeps successively through finer dyadic partitions. For $k\ge 0$ and $j=0,\dots ,2^k-1$, define the indicator functionIts L2 norm satisfiesAs $n$ tends to infinity so does its block index $k$, proving norm convergence to zero. However, every $x\in (0,1)$ belongs to exactly one of the half-open intervals in each block, so $f_n(x)=1$ infinitely often. For $k\ge 1$ there is also an interval in each block that does not contain $x$, so $f_n(x)=0$ infinitely often. Thus there is no pointwise convergence at any $x\in (0,1)$, let alone convergence almost everywhere.

Use the Fourier transform convention $\hat{F}(k)={\int }_{\mathbb{R}}F(x)e^{-ikx}dx$. First compute the Fourier coefficients of the periodized sum. Absolute integrability givesThus Fubini's theorem justifies termwise integration. For integer $m$, the change of variable $x=2\pi j+\tau$ and $e^{2\pi imj}=1$ giveReindexing a convergent periodization gives periodicity; the integral argument always gives its periodic representative almost everywhere. This proves the Fourier-coefficient content of the Poisson summation formula.

There is a genuine qualification to the printed pointwise claim: the periodization of an integrable function need not agree pointwise with its Fourier series under the stated hypotheses alone. Set $F(0)=1$ and $F(x)=0$ for $x\ne 0$. This is absolutely integrable, and on one closed period the periodization has only two possibly nonzero summands, so the series is uniformly convergent. Yet $f(0)=1$, while $\hat{F}(m)=0$ for every $m$. The claimed pointwise equality at zero is therefore false. This counterexample also works for Riemann integrability, not only for an almost-everywhere convention.

The coefficient identity gives a rigorous general replacement: the weighted Fejér sumsconverge to the periodization in the integral norm, and at every continuity point. Indeed they are its convolution with the normalized Fejér kernel; that kernel is nonnegative, has integral one, and has integral tending to zero outside every fixed neighbourhood of zero. For convergence in the L1 norm, split the convolution error into small translations, controlled by continuity of translation in the integral norm, and the remaining tail. The same split gives pointwise convergence at a continuity point. If the periodization is piecewise continuously differentiable, the ordinary Fourier series has the usual midpoint limit; if it is continuous and its coefficients are absolutely summable, the displayed unweighted formula holds pointwise. These are sufficient interpretations of the intended identity.

For the specified exponential example all the needed Fourier series convergence properties do hold. Split its Fourier transform at zero to obtainIts periodization is continuous, with uniformly geometrically bounded tails, and the sampled coefficients are absolutely summable. The absolutely summable coefficients define a uniformly convergent Fourier series with the computed coefficients. Its continuous sum equals the continuous periodization: their difference has every Fourier coefficient zero, hence is zero in the L2 norm by Fourier completeness, and continuity upgrades that to equality everywhere. Therefore its ordinary Poisson summation formula is valid. Evaluate it at zero:Thus the fully justified requested sum is

Yes; in fact the limit is uniformly continuous. Given $\epsilon >0$, uniform convergence provides one $N$ with ${\sup }_x|f_N(x)-f(x)|<\epsilon /3$. Uniform continuity of this particular $f_N$ provides $\delta >0$ such that $|x-y|<\delta$ implies $|f_N(x)-f_N(y)|<\epsilon /3$. The triangle inequality then givesThis same $\delta$ works at every point of $\mathbb{R}$, proving uniform continuity and therefore continuity. The use of one global $N$ is exactly what fails for mere pointwise convergence.

The integral is finite because a continuous function on a compact closed interval is bounded. Nonnegativity is immediate, and absolute homogeneity of a norm follows from $|cf(x)|=|c||f(x)|$. The pointwise triangle inequality givesFor positive definiteness, if $f(x_0)\ne 0$, continuity gives a relative interval of positive length on which $|f(x)|\ge |f(x_0)|/2$. Its integral is positive, including when $x_0$ is an endpoint. Thus $\Vert f\Vert =0$ implies $f=0$, and this is a norm on the given vector space.

For the given sequence,The last bound tends to zero as both indices tend to infinity, proving the Cauchy sequence property. Moreover, the sequence converges to the zero function in this norm:There is no conflict with the nonzero endpoint values: convergence in this norm does not imply pointwise convergence.

Uniform convergence means that for every $\epsilon >0$ there is an integer $N$, independent of $x$, such that $|F_n(x)-F(x)|<\epsilon$ for all $n\ge N$ and every $x\in (0,1)$. Equivalently ${\sup }_{0<x<1}|F_n(x)-F(x)|\to 0$. In pointwise convergence, the required $N$ may depend on the chosen $x$.

Uniform convergence of $f_n$ to $f$ on an interval $I$ means that for every $ϵ>0$ there is $N$ such that $|f_n(x)-f(x)|<ϵ$ for all $n\ge N$ and every $x\in I$. Equivalently ${\sup }_{x\in I}|f_n(x)-f(x)|\to 0$. The same $N$ must work throughout the interval, unlike pointwise convergence.

For completeness, continuity is preserved: choose $N$ with uniform error below $ϵ/3$, and use continuity of $f_N$ at a given $x$ to make $|f_N(y)-f_N(x)|<ϵ/3$. The triangle inequality then gives $|f(y)-f(x)|<ϵ$. Thus each function involved in the integral is continuous on the closed interval and Riemann integrable. The estimateproves convergence of the integrals by uniform convergence. When $a=b$ both integrals are zero.

Suppose the continuous functions converged pointwise to the rationality indicator. Each $F_N$ is closed, sincePointwise convergence implies that every point belongs to some $F_N$: its eventual values lie near either zero or one. Thus $\mathbb{R}={\bigcup }_N{F}_N$. By the Baire category theorem, some $F_N$ contains a nonempty open interval $I$. Choose a rational $r\in I$ and an irrational $s\in I$. For sufficiently large $n\ge N$, convergence gives $f_n(r)>2/3$ and $f_n(s)<1/3$. The intermediate value theorem then forces $f_n$ to take the value $1/2$ between them, at a point in $I\subseteq F_N$. This contradicts the defining condition of $F_N$. Therefore the rationality indicator is not a pointwise limit of continuous functions, the rationality indicator is not Baire class one result.

For each fixed $x>0$, continuity of the logarithm gives the pointwise convergence $f_n(x)\to \log x$. But at $x_n=1/n$, which belongs to $(0,1)$ for $n>1$,Consequently the uniform error cannot tend to zero. In fact $f_n(x)-\log x=\log (1+1/(nx))$ is unbounded as $x\downarrow 0$ for every fixed $n$. The pointwise limit is $\log x$, and convergence is not uniform.

The polynomial form of Runge theorem says that a function holomorphic on a neighbourhood of a compact set $K\subset \mathbb{C}$ with connected complement can be approximated uniformly on $K$ by polynomials.

For $n\ge 3$, takeThe complement is connected: the missing angular sector connects the inner disk to the exterior, and removing the single point zero from the inner disk does not disconnect it. On a neighbourhood of the annular sector choose an analytic logarithm with arguments between $-1/(2n)$ and $2\pi -1/(2n)$. Its exponential $\exp (3\log z/2)$ agrees with the prescribed function on the sector. On a disjoint small neighbourhood of zero prescribe the analytic function zero. These two neighbourhoods define a holomorphic function $g_n$ near $K_n$ with $g_n=f$ on $K_n$.

By Runge theorem, choose a polynomial $p_n$ with ${\sup }_{K_n}|p_n-g_n|<1/n$. Every fixed nonzero point of the disk eventually belongs to all $K_n$, including points on the positive real axis; zero belongs to every $K_n$. ConsequentlyEach polynomial restricts to an analytic function on $D$. The excluded sector lies immediately below the positive axis and shrinks with $n$, so the construction correctly preserves the chosen value on the branch cut. It gives pointwise convergence, not local uniform convergence across that cut. This is pointwise approximation by a Runge exhaustion.

Both functionals are finite on $X$: a continuous function on the compact metric space $[0,1]$ is bounded and integrable. Homogeneity follows from $|af|=|a||f|$, and the triangle inequality follows by integrating, or taking the supremum of, $|f+g|\le |f|+|g|$. Each is nonnegative. If $\Vert f{\Vert }_{\infty }=0$, then $f$ vanishes everywhere. If a continuous function $f$ is nonzero at some point, continuity supplies an interval of positive length on which $|f|$ is bounded below by a positive number; hence $\Vert f{\Vert }_1>0$. Thus both are norms.

For the nonnegative $f_n$, direct integration gives the L1 normTherefore $\boxed{f_n⟶0\text{ in }\Vert \cdot {\Vert }_1}$. On the other hand, differentiation places the maximum of $t^n(1-t)$ at $t=n/(n+1)$, giving the supremum normFor $t<1$, exponential decay implies $n{t}^n(1-t)\to 0$, while $f_n(1)=0$. Thus the pointwise convergence is to zero throughout $[0,1]$. Any convergence in the supremum norm would imply uniform convergence, hence the same pointwise limit zero, contradicting the displayed positive limiting norm. Therefore there is no convergence in the supremum norm.

Fix $ϵ>0$. For each $N$, the set $Q_N={\bigcap }_{n,m\ge N}\{x:|f_n(x)-f_m(x)|\le ϵ\}$ is closed, because each $f_n-f_m$ is continuous. At every $x$, pointwise convergence makes $(f_n(x))$ a Cauchy sequence, so $\mathbb{R}={\bigcup }_{N\ge 1}{Q}_N$. The Baire category theorem supplies an $N_0$ for which $Q_{N_0}$ contains a nonempty open interval $I$. For $x\in I$ and $n\ge N_0$, pass $m\to \infty$ in $|f_n(x)-f_m(x)|\le ϵ$ to obtainThis local uniform Cauchy control for pointwise convergent continuous functions allows both $I$ and $N_0$ to depend on $ϵ$.