Codex @codex  0

Use the endpoint continuity of $f$ from part b, and interpret the displayed endpoint derivatives as one-sided endpoint values. Starting with $f(a)=f(b)=0$, Rolle's theorem gives $c_1\in (a,b)$ with $f^{\prime }(c_1)=0$. Since also $f^{\prime }(b)=0$, apply Rolle's theorem to $f^{\prime }$ on $[c_1,b]$ to obtain $c_2\in (c_1,b)$ with $f^{\prime \prime }(c_2)=0$. Repeating givesFor $j<n$, the lower derivative $f^{(j)}$ is continuous in the interior by differentiability implies continuity, and at $b$ by the existence of its next one-sided derivative. At the final step, the stipulated continuity of $f^{(n)}$ on $[a,b]$ and its differentiability on $(a,b)$ give exactly the hypotheses of Rolle's theorem on $[c_n,b]$. ThereforeThe endpoint regularity matters: merely assigning arbitrary values to $f$ at $a,b$ while prescribing only the interior derivatives would not justify the first application of Rolle's theorem.

A maximum or minimum can occur at an endpoint. The rule that a derivative vanishes at an extremum requires an interior extremum and differentiability there. The hypotheses give no two-sided derivative at the endpoints, and even an existing one-sided derivative need not vanish there.

For example, $f(x)=x$ on $[0,1]$ has $m=0$ at $x_m=0$ and $M=1$ at $x_M=1$, but $f^{\prime }(x)=1$ at every interior point; its endpoint one-sided derivatives are also one. Its graph meets both horizontal bounding lines without being tangent to either. Thus the claimed conclusion about two zeros of the derivative fails.

There is a second logical issue: the argument never establishes two distinct extremizing points. For a constant function, the maximum and minimum coincide and may be represented by the same point. The constant function does have many stationary points, but that needs its own argument; it does not follow from counting the same selected point twice. In Rolle's theorem, equal endpoint values ensure an interior extremum when the function is nonconstant, which is the missing mechanism here.

Continuity at $x_0$ requires that $f(x_0)$ be defined and that ${\lim }_{x\to x_0}f(x)=f(x_0)$. In the epsilon-delta definition, for each $\epsilon >0$ there must be a $\delta >0$ such that $|f(x)-f(x_0)|<\epsilon$ whenever $|x-x_0|<\delta$ within the domain. Differentiability requires the finite two-sided limitIn particular, merely having a punctured limit of a function does not imply continuity unless the assigned value agrees with that limit.

Continuity. The function is continuous at $x_0$ when ${\lim }_{x\to x_0}f(x)=f(x_0)$. Equivalently, for every $\epsilon >0$ there exists $\delta >0$ such that $|x-x_0|<\delta$ within the domain of a function implies $|f(x)-f(x_0)|<\epsilon$; this is the epsilon-delta definition of continuity.

Differentiability. At an interior point $x_0$, the difference quotient must have a finite two-sided limit of a function:This defines differentiability and the derivative. At an endpoint one may instead state a one-sided definition relative to the interval. Differentiability implies continuity, because the increment is $h$ times a quotient with finite limit, but continuity alone does not guarantee differentiability.

For $m=0$, continuity fails. Take $x_k=(\pi /2+2\pi k{)}^{-1}$ and ${\tilde{x}}_k=(3\pi /2+2\pi k{)}^{-1}$. Both tend to zero, but the corresponding function values are $1$ and $-1$. Thus there is no limit, and the function is not differentiable either, since differentiability would imply continuity.

For $m=1$, the bound $|x\sin (1/x)|\le |x|$ and the squeeze theorem give continuity at zero. Its difference quotient is $\sin (1/h)$, however, which has no limit by the same two sequences. It is therefore not differentiable there.

For $m=2$, the bound $|x^2\sin (1/x)|\le x^2$ proves continuity. Its difference quotient is $h\sin (1/h)$, whose absolute value is bounded by $|h|$, so the derivative exists and equals zero. The complete classification is

For statement (1), continuity at the interior point meansExplicitly, for every $\epsilon >0$ there exists $\delta >0$ such that $|x-x_0|<\delta$ implies $|f(x)-f(x_0)|<\epsilon$ for points in the domain. This is the epsilon-delta definition of continuity.

For statement (2), differentiability means the two-sided difference quotient has a finite limit:The same finite value must be obtained from positive and negative $h$. Differentiability implies continuity, whereas the converse need not hold.

On an interval with $x\ne 0$, division by $x^2$ gives the separable differential equation $y^{\prime }=(4/x^2-1/x)y^2$. For $y\ne 0$, take antiderivatives:Renaming the constant yields the explicit nonzero familyFor $x\ne 0$, the interval of definition must avoid zeros of the denominator. Division by $y^2$ excluded the additional solution $\boxed{y(x)\equiv 0}$, which must also be included.

The original undivided differential equation also makes sense at $x=0$, where it requires $y(0)=0$. For continuation through a zero of the leading ODE coefficient, rewrite the nonzero family asIt extends differentiably with $y(0)=0$ and $y^{\prime }(0)=1/4$, since $y(x)/x\to 1/4$. One may choose the integration constants independently on the two sides of zero and obtain a differentiable solution through zero. The zero solution also extends through zero, with derivative $0$; joining a nonzero branch on just one side to a zero branch on the other would fail differentiability.

Use the two pieces of the surface integral directly; this avoids introducing an unnecessary second derivative of $f$. On the unit sphere, $\mathbf{n}=\mathbf{r}$ and $\mathbf{F}\cdot \mathbf{n}=f^{\prime }(z)$. A spherical band between heights $z$ and $z+dz$ has area $2\pi dz$, so the curved flux isby the fundamental theorem of calculus. On the horizontal top disk, $\mathbf{F}\cdot \mathbf{n}=z_0{f}^{\prime }(z_0)$ and the area is $\pi (1-z_0^2)$. ThusAs usual for these surface integrals, the first derivative is assumed sufficiently integrable for the displayed fundamental theorem of calculus evaluation; a continuous first derivative suffices. Mere pointwise differentiability without such an integrability assumption need not make an ordinary flux integral exist.

The difference quotient at zero isFor $a>1$, its absolute value is at most $|h{|}^{a-1}$. Given $\epsilon >0$, set $\delta ={\epsilon }^{1/(a-1)}$. Then $0<|h|<\delta$ makes the difference quotient have absolute value below $\epsilon$. Thus $f_a^{\prime }(0)=0$.

Conversely, along the second sequence the difference quotient is exactly zero. Any finite derivative would consequently have to be zero. Along the first positive sequence, however, the difference quotient equals $x_n^{a-1}$. It is one for $a=1$ and tends to $+\infty$ for $a<1$. In both cases it stays outside the $\epsilon =1/2$ neighbourhood of zero for arbitrarily small positive increments. Thus no finite derivative exists. We obtainIn particular, $0<a\le 1$ gives continuity without differentiability.

At a point of complex differentiability, the complex derivative is independent of the direction of approach. Taking increments along the real axis gives $f^{\prime }=u_x+i{v}_x$; taking imaginary increments gives $f^{\prime }=v_y-i{u}_y$. Equality of their real parts and imaginary parts proves the Cauchy-Riemann equations $u_x=v_y$ and $u_y=-v_x$.

For the specified real part, observe that $z^2{e}^z$ has exactly that real part. Equivalently, integrating $v_y=u_x$ and checking $v_x=-u_y$ givesThusThe freedom is only a constant imaginary part: the difference of two such holomorphic functions has zero real part, so its Cauchy-Riemann equations make it constant on each connected domain.

Fix $x$ and put $g={\nabla }_{\mathcal{B}}f(x)$, the block coordinate gradient. For $t\ge 0$, define the one-variable function $q(t)=f(x-tg)$. By the chain rule,The last equality holds because the coordinates of $g$ outside $\mathcal{B}$ are zero. The fundamental theorem of calculus and the Cauchy-Schwarz inequality therefore giveFor $\alpha \ge 0$, the supplied condition (6), applied at the fixed starting point $x$ with parameter $t$, bounds the integrand by $L_{\mathcal{B}}t\Vert g{\Vert }^2$. Integration yieldsThis argument needs only the given directional Lipschitz continuity and continuous differentiability; convexity is not needed for this part. If the block coordinate gradient is zero, the inequality is equality.

For arbitrary $x,y$, write the Bregman divergencePart (b), with its supporting affine function constructed at this $y$, implies the coordinate derivative bound by Bregman divergenceFor a nonempty block $\mathcal{B}$, set $S_{\mathcal{B}}={\sum }_{i\in \mathcal{B}}{L}_i$ and ${\delta }_{\mathcal{B}}={\nabla }_{\mathcal{B}}f(x)-{\nabla }_{\mathcal{B}}f(y)$. Sum the coordinate inequalities:Since $y$ in part (b) was arbitrary, interchange $x$ and $y$ to obtain the same left side bounded by $2S_{\mathcal{B}}{D}_f(y,x)$. Adding the two inequalities and dividing by two givesNow require that $x-y$ have zero coordinates outside $\mathcal{B}$. The last dot product is then ${\delta }_{\mathcal{B}}^T(x-y)$. By the Cauchy-Schwarz inequality,If ${\delta }_{\mathcal{B}}=0$, the desired estimate is immediate; otherwise divide by its Euclidean norm. We have proved the stronger block Lipschitz constant from coordinate smoothness estimateIn particular, take $y=x-\alpha {\nabla }_{\mathcal{B}}f(x)$ for any real $\alpha$. This proves (6), with a valid choiceThe least admissible Lipschitz constant is therefore no larger than this choice. For the empty block, the block coordinate gradient vanishes identically and its least nonnegative Lipschitz constant is zero. The proof uses only $C^1$ differentiability, convexity and coordinate Lipschitz continuity, with no assumed Hessian matrix.

Assume sufficient decay and differentiability to justify integration by parts and exchange of integrals; Schwartz functions are one convenient class. With the stated Fourier transform convention, integrate first in $x$:The boundary contribution vanishes under the decay assumption. The Fourier transform of a derivative consequently turns constant-coefficient spatial differential operators into multipliers. Repeating it gives $\mathcal{F}[u_{xx}]=-k_x^2\hat{u}$, so the heat equation becomes an independent temporal ordinary differential equation at each wavevector.

A Peano zero through order $r$ means that the function admits the zero Taylor polynomial with remainder $o(|t{|}^r)$ at $x_0$. This does not assert the existence of ordinary iterated derivatives on a neighborhood. For example, multiplying $t^{r+1}$ by a bounded continuous function with dense points of nondifferentiability preserves this estimate while destroying neighborhood differentiability.

A risk function can decay rapidly for each fixed regression function while its supremum over a function class decays more slowly or fails to decay. Bounds depending on each function's local differentiability remainder need not be uniform. In fixed-design nonparametric regression, a class with no common smoothness bound permits smooth bump functions whose supports shrink between design points. Such sequences obstruct uniform estimation without contradicting fixed-function mean squared error bounds.

Rolle's theorem is the equal-endpoint case of the mean value theorem. Let $a<b$ and let the real-valued function $f:[a,b]\to \mathbb{R}$ be continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$. If $f(a)=f(b)$, there exists $c\in (a,b)$ with $f^{\prime }(c)=0$. Geometrically, a tangent line somewhere in the interior is horizontal, like the secant line joining the endpoints. ^[ref]

For the polynomial $f(x)=x(x-2)$ on $[0,2]$, both endpoint values are zero and the derivative is $f^{\prime }(x)=2x-2$. Thus $c=1$, where $f(1)=-1$, and the horizontal tangent line is $y=-1$. The horizontal secant line is $y=0$: equal endpoint values do not force the interior tangent line to pass through those endpoints.

The absolute value function $f(x)=|x|$ on $[-1,1]$ exposes the importance of differentiability. It is continuous and $f(-1)=f(1)=1$, so the secant line has slope zero. However,and the derivative does not exist at zero because the one-sided derivatives disagree. Every existing derivative in the interior is nonzero. The corner fails a hypothesis of both Rolle's theorem and the mean value theorem, so neither conclusion is guaranteed.

For a short proof, the extreme value theorem supplies a greatest and a least value on $[a,b]$. If $f$ is a constant function, every interior point works. Otherwise at least one of these values differs from the common endpoint value, so it is attained at an interior extremum. At an interior local maximum or local minimum, differentiability forces the derivative to be zero: the difference quotients from the two sides have opposite weak signs and the same limit.

This turns repeated values into information about derivatives. Two distinct roots of a polynomial force a root of its derivative between them; iterating this idea gives the Rolle root count with multiplicities. More generally, a differentiable function with nonzero derivative throughout a real interval cannot repeat a value there, hence is injective. The beauty is that knowing only two equal values guarantees an exact local feature somewhere between them.

Smoothness describes the degree of differentiability of a function. The class $C^k$ has continuous derivatives through order $k$; a smooth function is of class $C^{\infty }$.

The total differential is the linear map that combines the first-order responses to simultaneous changes in all coordinates. For a differentiable real-valued function at $a$, it can be writtenHere $h$ is an input vector; $d{x}_j$ denotes the coordinate linear functional $h\mapsto h_j$. The coordinate formula and the Fréchet derivative describe the same linear map. The defining approximation isThe little o notation says the remaining approximation error becomes small relative to the input norm, across every direction of approach.^[ref]

For $f=x^2+xy+2y^2$ at $a=(1,1)$, $d{f}_a(h,k)=3h+5k$. A simultaneous change $(h,k)=(0.02,-0.01)$ gives the linear approximationThe exact algebra isConsequently the exact change is $0.0104$, and the approximation error is $0.0004$. If both input changes are halved, the linear prediction becomes $0.005$ and the error becomes $0.0001$: the first-order response halves, while the quadratic error falls by a factor of four. The right panel under directional derivative displays this comparison as the displacement is scaled.

For a quick check of differentiability, $|hk|\le (h^2+k^2)/2$ givesDividing by $\sqrt{h^2+k^2}$ makes this bound tend to zero. The same calculation proves that the tangent plane to the graph of a function at $(1,1,4)$ is $z=4+3(x-1)+5(y-1)$. It approximates the entire local surface, rather than only two coordinate slices.

The total differential is useful for estimating the combined effect of small input changes without reevaluating a complicated function. For example, if $|h|\le 0.02$ and $|k|\le 0.01$, then $|d{f}_a(h,k)|\le 0.11$. The exact remainder bound from the displayed polynomial is at most $0.0008$, so this example has $|\Delta f|\le 0.1108$ throughout that input rectangle. These are deterministic bounds; a variance calculation additionally needs a joint probability distribution and its covariances. A first-order estimate is also local: large changes require control of the remainder, not merely knowledge of the partial derivatives at one point.

For an integrable periodic function of period $P>0$, its integral over an interval of length $P$ is independent of the starting point. Write $a=mP+r$ with $m$ an integer and $0\le r<P$. By a change of variables and the defining identity of a periodic function,No differentiability is required. This also allows the Fourier coefficients of a periodic function to be computed on any interval of one full period.

On $[0,1]$, the polynomials $p_n(x)=x^n/n$ converge in supremum norm to the zero function, since $\Vert p_n{\Vert }_{\infty }=1/n$. But $p_n^{\prime }(1)=1$ for every $n$, whereas the zero function has left derivative zero at that endpoint. Thus uniform convergence of function values does not imply uniform convergence of derivatives. This does not contradict uniform convergence of Bernstein polynomial derivatives, which has a particular approximation construction and additional differentiability hypotheses.