Articles of real analysis

Derivative change sign

If the function $f:\mathbb{R} \to \mathbb{R}$, $f$ has an extremum at the point $x$, and $f$ is differentiable in some neighborhood of $x$. Is it right that the derivative changes sign when passing through $x?$

Is there a bijection between $(0,1)$ and $(0,infinity)$

This question already has an answer here: How to define a bijection between $(0,1)$ and $(0,1]$? 7 answers

In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous

In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous The first part of the question says to prove the “reverse triangle inequality” which is $\lvert u\rvert -\lvert v\rvert \le \lvert u-v\rvert$ I sense that might be a clue. Normally I’d start from the definitions, but I’m […]

Showing continuity of a function that depends on another continuous function.

Question: please help me pointing out the errors of my proof (I’m sure there are some). The proof is structured in cases (two cases with each two subcases) and I think that some may be correct but others won’t. So instead of just writing a different proof I would be thankful if you could point […]

Lusin's Theorem when $m(E)=+\infty$. Proof Verification.

Lusin’s Theorem. Let $f$ be a real-valued measurable function on $E$. Then for each $\varepsilon>0$, there is a continuous function $g$ on $\mathbb{R}$ and a closed set $F$ contained in $E$ for which $f=g$ on $F$ and $m(E\sim F)<\varepsilon$. Let us suppose we have proved the case when the measure of $E$ is finite. $m(E)= […]

Function that is not uniformly differentiable

Define $f: A \to \mathbb R$ ($f$ differentiable)to be uniformly differentiable if and only if for $\varepsilon >0$ there exists $\delta >0$ such that $$ |h| < \delta \implies \left| {f(x + h) – f(x) \over h} -f'(x) \right | < \varepsilon$$ I am looking for an example of $f$ that is not uniformly differentiable. My […]

Simplified l'Hospital: please can you check my proof

I tried to show: If $f,g: [a,b) \to \mathbb R$ are differentiable and $\lim_{x \to a^+}f(x) = \lim_{x \to a^+} g(x) = 0 $ and $\lim_{x \to a^+} {f'(x) \over g'(x)} = L$. Also assume that $f’$ and $g’$ are continuous. Then $\lim_{x \to a^+} {f(x) \over g(x)} = L$. Please can you check my […]

Question About Concave Functions

It easy to prove that no non-constant positive concave function exists (for example by integrating: $ u” \leq 0 \to u’ \leq c \to u \leq cx+c_2 $ and since $u>0$ , we obviously get a contradiction. Can this result be generalized to $ \mathbb{R}^2 $ and the laplacian? Is there an easy way to […]

If $(F_t)_t$ is a filtration, $T$ is a stopping time and $Y$ is $F_T$-measurable, then $1_{\left\{T=s\right\}}Y$ is $F_s$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq[0,\infty)$ $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $\tau$ be a $\mathbb F$-stopping time $\mathcal F_\tau:=\left\{A\in\mathcal A:A\cap\left\{\tau\le t\right\}\in\mathcal F_t\;\text{for all }t\in I\right\}$ $Y:(\Omega,\mathcal F_\tau)\to\left(\mathbb R,\mathcal B\left(\mathbb R\right)\right)$ be a random variable $s\in I$ and $Z:=1_{\left\{\tau=s\right\}}Y$ Can we show, that $Z$ is $\mathcal F_s$-measurable? Clearly, we’ve […]

Passing limit through derivative!

In the book “General topology” of Lipschutz, exercise 26 at page 222 says that: if $(f_n)$ be a sequence of real value differentiable funtions on $[a, b]$ which converge uniformly to $g$ then $$\lim_{n\to \infty} \frac{d}{dx}f_n(x)= \frac{d}{dx}\lim_{n\to \infty}f_n(x).$$ But I find that the sequence $$f_n(x)=\frac{1}{n}\cos(nx), \ \ x\in [0, 1]$$ satisfying $f_n$ converge uniformly to […]