Intereting Posts

Explanation Borel set
Tail bound on the sum of independent (non-identical) geometric random variables
What do ideles and adeles look like?
Yamabe's equation
Operator whose spectrum is given compact set
Prove that $f_A (x) = d({\{x}\}, A)$, is continuous.
How many primes does Euclid's proof account for?
Sturm-Liouville Questions
Sherman-Morrison formula and a sum of outer products
Structure Theorem for abelian torsion groups that are not finitely generated
Prove that $N$ is normal
Evaluating $\sum\limits_{n=0}^{20} \frac{(-1)^{n}2^{n+1}}{3^{n}},$
Finding Divisibility of Sequence of Numbers Generated Recursively
Why doesn't this work imply that there are countably many subsets of the naturals?
Prove that the only prime triple is 3, 5, 7

If you know the solution for this exercise, I would appreciate a **HINT**:

Let $f:U\longrightarrow\mathbb{R}$ a function defined in an open subset $U$ of $\mathbb{R}^m$. Given $p\in U$, suppose that, for every path $\lambda:(-\epsilon,\epsilon)\longrightarrow U$, with $\lambda(0)=p$, that has a velocity vector $v=\lambda ‘(0)$ at $t=0$, the composed path $f\circ\lambda:(-\epsilon,\epsilon)\longrightarrow\mathbb{R}$ also has a velocity vector $(f\circ\lambda)'(0)=Tv$, where $T:\mathbb{R}^n\longrightarrow\mathbb{R}$ is linear. Prove that, under these conditions, $f$ is differentiable at $p$.

[ NOTE: I’ve been thinking about it for a while now. In doing so, I came up with this other question (poorly formulated, but please see my comments on the second answer): Always a differentiable path through a convergent sequence of points in $\mathbb{R}^n$? ]

- A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$
- Absolute value in trigonometric substitutions
- Do the infinite series converge
- Deriving Taylor series without applying Taylor's theorem.
- Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$
- How does one approximate $\cos(58^\circ)$ to four decimal places accuracy using Taylor's theorem?

- Uniform Convergence verification for Sequence of functions - NBHM
- Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$
- Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$
- Convergence of $\sum \limits_{n=1}^{\infty}\sin(n^k)/n$
- Prove that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$
- Gaps in my proof of the Arzela-Ascoli Theorem - help and expertise greatly appreciated for an alternate formulation.
- Is this space complete?
- Proving that $\lim\limits_{x\to 0}\frac{f(x)}{g(x)}=L$ implies $\lim\limits_{r\to 0}\frac {\int_R f(ry)h(y)\,dy}{\int_R g(ry)h(y)dy}=L$
- Actuarial : “ Amortization - mortage”
- How to prove that $\lim(\underset{k\rightarrow\infty}{\lim}(\cos(|n!\pi x|)^{2k}))=\chi_\mathbb{Q}$

The chain of comments became too long, so I’m switching to the answer box.

- We may assume that $T=0$ by subtracting $Tx$ from our function.
- Suppose $f$ is not differentiable. Pick a sequence $v_k\to 0$ such that $|f(v_k)|\ge \epsilon |v_k|$.
- Passing to a subsequence, make sure that $v_k/|v_k|\to u$, and also that $|v_{k+1}|\le \frac{1}{2}|v_k|$. (We don’t want the sequence to jump back and forth.)
- Connect the points by line segments. Parametrize this piecewise-linear curve $\lambda$ by arclength, which is finite.
- The length of $\lambda$ between $0$ and $v_k$ is bounded by $4|v_k|$ or some such multiple.
- $\lambda$ has a one-sided derivative when it reaches $0$. Extend it to get two-sided derivative at that point (apparently, this is all the statement requires; the entire path need not be smooth)
- By assumption, $|f(\lambda(t))|/|t|\to 0$ as $t\to\infty$. This contradicts 2&5.

- When is the derivative of an inverse function equal to the reciprocal of the derivative?
- Explanation of Lagrange Interpolating Polynomial
- Can we extend the definition of a homomorphism to binary relations?
- How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without using l'Hopital's rule nor any series expansion?
- How can I convert fifth order polynomial to a linear equation using logs?
- Hypergeometric function integral representation
- Another sum involving binomial coefficients.
- Dot product in coordinates
- Is the closure of $ X \cap Y$ equal to $\bar{X} \cap \bar{Y}$?
- If $\sum a_n^2 n^2$ converges then $\sum |a_n|$ converges
- Prove $f_n(x)=(1-x/n)^n$ converges uniformly on non-negative reals
- How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$
- How to prove(or disprove) $\begin{vmatrix} A&B\\ B&A \end{vmatrix}=|A^2-B^2|$
- How do I evaluate this limit: $\lim_{n\to+\infty}\sum_{k=1}^{n} \frac{1}{k(k+1)\cdots(k+m)}$?
- Area of intersection between two circles