Newbetuts

Differentiable bijection $f:\mathbb{R} \to \mathbb{R}$ with nonzero derivative whose inverse is not differentiable

Show that $\limsup \frac{\epsilon_1+\cdots +\epsilon_n}{n}\geq 0$？

If $|g(a)-g(b)| \leq |f(a)-f(b)|,$ for every $a,b$, and $f$ is a Darboux function, then $g$ is a Darboux function.

prove the Riemann-Lebesgue lemma: $\int^b_af(x)\cos(nx)dx\rightarrow 0$ as $n\rightarrow \infty$ for any regulated function $f$

Computing the limit $\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$ for fixed $n \in \mathbb{N}$

Calculus of Variations and Lagrange Multipliers

Is $f(2x)/f(x)$ nonincreasing for concave functions with $f(0)=0$?

Uniformly distributed rationals

A limit with integral

Norm on a Hölder's space

Differentiability of Norms

Integral-Summation inequality.

Stuck on existence proofs involving measurability and simple functions

Are functions that map dense sets to dense sets continuous?

Show that every interval is a Borel set

If $a_1,a_2,\dotsc,a_n>0 $, then $\lim\limits_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+\dotsb+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n$

Continuity of the inverse $f^{-1}$ at $f(x)$ when $f$ is bijective and continuous at $x$.

New posts in real-analysis

Confusion with the narrow and weak* convergence of measures

Everywhere Super Dense Subset of $\mathbb{R}$

Counterintuitive Charecterization of Openness in Euclidian Topology

Differentiable bijection $f:\mathbb{R} \to \mathbb{R}$ with nonzero derivative whose inverse is not differentiable

Show that $\limsup \frac{\epsilon_1+\cdots +\epsilon_n}{n}\geq 0$？

If $|g(a)-g(b)| \leq |f(a)-f(b)|,$ for every $a,b$, and $f$ is a Darboux function, then $g$ is a Darboux function.

prove the Riemann-Lebesgue lemma: $\int^b_af(x)\cos(nx)dx\rightarrow 0$ as $n\rightarrow \infty$ for any regulated function $f$

Computing the limit $\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$ for fixed $n \in \mathbb{N}$

Calculus of Variations and Lagrange Multipliers

Is $f(2x)/f(x)$ nonincreasing for concave functions with $f(0)=0$?

Uniformly distributed rationals

A limit with integral

Norm on a Hölder's space

Differentiability of Norms

Integral-Summation inequality.

Stuck on existence proofs involving measurability and simple functions

Are functions that map dense sets to dense sets continuous?

Show that every interval is a Borel set

If $a_1,a_2,\dotsc,a_n>0 $, then $\lim\limits_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+\dotsb+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n$

Continuity of the inverse $f^{-1}$ at $f(x)$ when $f$ is bijective and continuous at $x$.