# Tagged Questions

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### A Tricky Weak Derivatives question

I recently came across the following statement and am having trouble proving it correct. I wonder if you could help. Given a weak derivative, $x'$, there exists an absolutely continuous ...
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### Differentiability of product/composition of function

How will be the product and composition of two functions, where one is differentiable and another is just continuous, behave?I mean to say, if the product or composition is differentiable, then what ...
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### Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
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### Mean Value related problem.

I'm working on a function $f : \left[a,a+h\right] \rightarrow \mathbb{R}$. I know that $f$ satisfies the conditions of the Mean Value Theorem thus I have $\theta \in \left(0,1\right)$ such that ...
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### Proof using Fermat's Theorem on stationary points?

It seems intuitive that, if a function differentiable on [a,b] is such that f'(a) < 0 < f'(b) then there exists some c in the open interval (a,b) such that f'(c)=0, but I can't prove it ...
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### Prove a functional to be differentiable

Consider the functional $I:C[a,b]\rightarrow\mathbb{R}$ given by $$I(x)=\int_{b}^{a}x(t)dt.$$ Prove that $I$ is differentiable and find its derivative at $x_0\in C[a,b]$. The answer just says that it ...
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### Example: Differentiable, NOT locally lipschitz?

Everybody Good Evening, I'm wondering: Is being differentiable (Gateaux or Frechet) at ONE point enough for being locally lipschitz? If not, can you provide a counterexample? Thx in advance!
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### Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$\lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t}$$ exists then $\psi$ ...
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### What is the Frechet derivative of $(u^+)^q$?

I know that if we define $E[u]=\int_\Omega u^+dx$, where $\Omega$ is compact in $R^n$ and $u\in H_0^1(\Omega)$, $u^+:=\max\{u,0\}$, then $E[u]$ is not Frechet differentiable. However, if now I define ...
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### Differentiability in $\mathbb R^n$

Let $U\in \mathbb{R}^n$ be open, and let $f:U\to \mathbb{R}^m$, and let $a\in U$. Let $\|\cdot\|'$ be a norm on $\mathbb{R}^n$, and let $\|\cdot\|''$ be a norm on $\mathbb{R}^m$. Prove that $f$ is ...
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### Frechet Derivative: Why bounded (linear) operator?

Why do we want the frechet derivative to be a bounded linear operator? (This meant more as a collecting ideas - I know bounded operators behave fine but that would exclude alot of examples such as the ...
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### Implicit function theorem and derivative (proof of splitting lemma)

I have this theorem with a part of the proof: $\quad$ Let $V$ be a Hilbert space, $U$ an open neighborhood of $u\in V$, and let $\varphi\in C^2(U,\mathbf R)$. Define implicity the linear operator ...
For example, I have a functional $$J(f)=\int \frac{f(x)}{1+x^2}dx.$$ How to calculate $\frac{\partial J}{\partial f(x)}$? Does it equal to $\int \frac{1}{1+x^2}dx$? It seems that the question is ...