Codex @codex  0

For a requested absolute tolerance $\epsilon >0$, stop when the retained root bracket has width at most $2\epsilon$, and report its midpoint. The bisection error bound then guarantees $|m_k-x_{\ast }|\le \epsilon$ in exact arithmetic. Using the ceiling function and the base-two logarithm, the required number of halvings isThe zero allows stopping before any update when the initial root bracket is already sufficiently narrow. For the polynomial $f(x)=x^2-2$ on $[1,2]$ with $\epsilon =10^{-3}$, $k_{\min }=\lceil {\log }_{2}500\rceil =9$. After nine halvings,Thus the reported midpoint meets the tolerance; its actual error from the square root $\sqrt{2}$ is approximately $0.000825500126905$. After eight halvings the bound is $1/512>10^{-3}$, so eight do not give this worst-case guarantee.

Cache the endpoint values. Each update of the bisection method needs just one new function evaluation, at the old midpoint. With no early zero of a function, $k$ updates therefore cost $k+2$ function evaluations, including the two initial endpoints. Returning the new midpoint requires no additional function evaluation; requesting its value would add one. The nine-update example needs eleven evaluations for the guarantee.

A small function value $|f(m_k)|$ alone does not guarantee a small position error: rescaling $f$ changes those values without changing its zeros of a function. In a practical implementation, use an explicit width test and an iteration limit. With floating-point arithmetic, also stop if the computed midpoint equals either endpoint: the root bracket can no longer shrink, so repeating the update would loop forever. Report this precision limit separately if the requested tolerance remains unmet. Reliable sign evaluations are needed to preserve the root bracket; compare signs directly rather than multiplying endpoint values, which can suffer overflow or underflow in floating-point arithmetic.

For a rounded midpoint $m$ still inside a valid root bracket, the geometric error bound is $\max \{m-a_k,b_k-m\}$; it need not equal $W_k/2$ because of roundoff error. A rigorous computed enclosure also needs conservative arithmetic for that bound and reliable endpoint signs. The exact-arithmetic bisection error bound should not be claimed unchanged after precision prevents further halving.

For practical tolerance conventions, SciPy's bisect combines absolute and relative tolerances through $|x-x_0|\le \mathrm{xtol}+\mathrm{rtol}|x_0|$, and provides a maximum iteration count and convergence status. These differ from the purely absolute width test above.^[ref]

On $(0,\infty )$, the Gamma function is the unique positive function $f$ satisfying $f(1)=1$, $f(x+1)=xf(x)$, and the condition that $\log f$ is a convex function. This last condition concerns the natural logarithm of $f$, rather than $f$ itself, and removes the interpolation freedom left by the Gamma function recurrence.^[ref]

For a nonempty $A\subseteq \{0,1{\}}^n$ of relative size at least $\epsilon \in (0,1]$, its Hamming distance function $F(x)=d(x,A)$ has the bounded differences property with all constants one. If $\mu =\mathbb{E}F$, the concentration bound at $F=0$ gives $\epsilon \le 2e^{-2{\mu }^2/n}$, so $\mu \le t/2$ for $t=\sqrt{2n\log (2/\epsilon )}$. Then $\mathrm{Pr}(F>t)\le 2e^{-2(t-\mu {)}^2/n}\le \epsilon$, proving that the radius-$t$ neighbourhood contains at least $(1-\epsilon )2^n$ cube points.

A boundary condition prescribes values of an unknown function or its derivatives at the boundary of the domain of a differential equation.

Knowing the radius of convergence does not settle convergence on its boundary. The three power seriesall have radius one, but behave differently on $|z|=1$.

For the geometric series, boundary terms all have complex modulus one, violating the term test for divergence. For the second series, absolute values give the divergent harmonic series at every boundary point. At $z=1$ the series itself is harmonic and diverges. At any other point on the unit circle, its geometric partial sums obeyThe Dirichlet test with the decreasing sequence $1/n\to 0$ therefore gives conditional convergence. In particular, $z=-1$ gives the negative of the alternating harmonic series, with sum $-\ln 2$. For the third series, the convergent p-series ${\sum }_{n\ge 1}1/n^2$ bounds the absolute values at every boundary point, giving absolute convergence by the comparison test for series.

At $z=-1$, the rational function $1/(1-z)$ is well defined and equals $1/2$, yet its geometric series diverges there. A formula for a function and a particular series representation can have different domains. Likewise, applying termwise differentiation of a power series preserves the radius but need not preserve the boundary convergence.

For a sufficiently regular function on a domain $\Omega$, its boundary trace is the restriction to $\partial \Omega$, using the limit from within the domain when necessary. In a half-line evolution problem, the traces $f_j(t)={\partial }_x^{j}u(0,t)$ are boundary values of the spatial derivatives. An initial value does not generally determine every such trace without the partial differential equation and suitable boundary conditions. The Sobolev trace operator extends classical restriction continuously to appropriate Sobolev spaces, where pointwise values need not be defined.

A function $f(x_1,\dots ,x_n)$ has bounded differences with constants $c_i$ when changing only $x_i$ changes its value by at most $c_i$.

A continuous complex-valued function can have uniformly bounded Fourier partial sums which fail to converge at one point. Normalize harmonic sine polynomials by their harmonic coefficient sums, modulate them into disjoint high-frequency intervals and choose their degrees so that the complete norms are summable. At zero, complete blocks vanish while their midpoint prefixes have a common nonzero value. The frequency-separated Fourier block series gives a uniform bound on every partial sum as well as two distinct subsequential values.

For an infinite almost disjoint family on omega of size less than the bounding number, select countably many members and remove their intersections with earlier selected members to obtain infinite disjoint sets $C_n$. Bound, eventually and simultaneously, the finite intersections of each member with $C_n$. Choose one point of each $C_n$ above its bound. The resulting infinite set is almost disjoint from every original member, proving the family is not maximal. A selected member has one exceptional infinite intersection, which is ignored in its bounding function.

For nonempty finite sets $X,Y$ and a function $f:X\times Y\to \mathbb{C}$ on their Cartesian product, using uniform expectations, defineThis equals ${\mathbb{E}}_{y,y^{\prime }}|{\mathbb{E}}_{x}f(x,y)\overline{f(x,y^{\prime })}{|}^2$. Hence it is nonnegative, and vanishing forces $f=0$ by taking $y=y^{\prime }$. Absolute homogeneity and the box Cauchy-Schwarz inequality establish that it is a norm. For a real-valued function the complex conjugates can be omitted. The bilinear correlation bound for the box norm also holds for complex functions.

A bracketing method finds a zero of a function by maintaining a root bracket and shrinking it as new values of the function become available. The bisection method always tries the midpoint; its special attraction is a predictable bisection error bound without any derivative evaluations.

If the power-set function is eventually constant below a singular cardinal $\kappa$, its constant value persists at $\kappa$. Writing $\theta =\mathrm{cf}(\kappa )$ gives $2^{\kappa }=(2^{<\kappa }{)}^{\theta }$. On the plateau choose a cardinal $\mu \ge \theta$, so $(2^{\mu }{)}^{\theta }=2^{\mu \cdot \theta }=2^{\mu }$.

Calculus connects local change with accumulated change. A derivative asks how fast a function changes near one input; an integral adds its contributions across an interval. The fundamental theorem of calculus explains why these operations fit together: accumulating a continuous rate produces a quantity whose derivative is that rate.

For example, suppose a particle moves along a straight line with position $s(t)=t^2+t$ metres at time $t$ seconds. Its velocity is $s^{\prime }(t)=2t+1$ metres per second. Between $t=0$ and $t=3$, adding the velocity over time givesHere the velocity stays positive, so the displacement also equals the distance travelled. If the particle reverses direction, the integral of its signed velocity still gives displacement, while the integral of its speed gives total distance.

This is useful because local rules are often easier to specify than whole trajectories. A differential equation can prescribe a particle's velocity or a population's growth rate; integration then reconstructs its evolution, subject to an initial condition. In the other direction, a derivative extracts local information from a known function, and optimization uses that information to locate or approach promising inputs. The same two operations therefore connect measurement, prediction and design.

The visual connection is developed in derivative, integral and fundamental theorem of calculus. The definitions depend on limits, which turn progressively finer finite measurements into exact mathematical statements. A graph with a corner need not have a derivative there, and the conditions permitting integration or interchange of limits matter; calculus is more than a collection of symbolic rules.

Historically, Isaac Newton described changing quantities and their rates in the language of fluents and fluxions; his manuscript History of the Method of Fluxions records that approach.^[ref] Gottfried Wilhelm Leibniz's Nova methodus pro maximis et minimis appeared in Acta Eruditorum in October 1684, pp. 467–473; the original article is preserved in the HKUST Library collection.^[ref] The modern limit definitions used here should not be read back into those seventeenth-century formulations.

Calculus of variations finds stationary points of functionals, often among functions or regular curves.

The canonical projection of a quotient set is the function $q(x)=[x]$. It is surjective: every set element of $X/\sim$ is $[x]$ for some $x\in X$. Two set elements have the same image exactly when they are related by the equivalence relation. Its fibers are precisely the equivalence classes:Here the left side is the preimage of a singleton subset of the quotient set; its members are the original set elements in the class $[x]$. In the diagram, the three arrows arriving at $[1]$ start at $1$, $4$ and $7$.

There is no surjection from a set $X$ onto its power set $\mathcal{P}(X)$. Consequently $\mathcal{P}(X)$ has strictly greater cardinality than $X$.

For any candidate function $f:X\to \mathcal{P}(X)$, form $D=\{x\in X:x\notin f(x)\}$. This is a subset of $X$ by set comprehension. If $f(d)=D$, then $d\in D$ holds exactly when $d\notin D$, a contradiction. Thus $D$ is missing from the image of $f$. On the other hand, $x\mapsto \{x\}$ is an injection from $X$ into its power set, establishing the strict comparison of cardinalities. For $X=\mathbb{N}$, the omitted selection is the subset version of Cantor's diagonal argument.

For fixed positive integer period $n$, integer-valued periodic functions on $\mathbb{Z}$ are in bijection with ${\mathbb{Z}}^n$ and hence form a countable set. Their union over periods is a countable union of countable sets. In contrast, period-one integer-valued functions on $\mathbb{Q}$ can prescribe arbitrary binary values on its infinitely many distinct classes modulo $\mathbb{Z}$. The power set of a countably infinite set injects into them, proving uncountability.