New posts in real-analysis

Prove that $\int_a^bxf(x)dx\geq\frac{b+a}{2}\int_a^bf(x)dx$

real-analysis integration integral-inequality

If $f : [a,b]\to\Bbb R$ is continuous, are there $x_1,x_2\in (a,b)$ such that $\tfrac{f(b)-f(a)}{b-a} = \tfrac{f(x_1)-f(x_2)}{x_1-x_2}$?

real-analysis continuity

Existence of a pointwise convergent subsequence

real-analysis general-topology

Limit of $\int_0^1\left(\frac {2}{\sqrt {(1-t^2)(1-xt^2)}}-\frac{x}{1-xt}\right)\,dt$ as $x\to 1^{-}$

real-analysis integration limits

An inequality regarding power of two real numbers

real-analysis calculus inequality exponential-function

Understanding that $A = \{x\in \ell_2: |x_n| \leq \frac{1}{n}, n = 1,2,...\}$ is compact in $l_2$.

real-analysis sequences-and-series compactness

Show by contradiction that $a^n\to 0$

Increasing functions are Baire one

Loomis and Sternberg Problem 1.58

real-analysis calculus linear-algebra vector-spaces linear-transformations

Some basics of Sobolev spaces

real-analysis functional-analysis sobolev-spaces

Derivative of a function is odd prove the function is even.

Strong convergence of operators

real-analysis analysis functional-analysis operator-theory banach-spaces

If $f:\left[a,b \right]\rightarrow R$ is integrable and ${N}_{f}=\{x\in\left[a,b \right]: f(x)=0\}$ is dense, so $\int_{a}^{b}f(x)dx=0$

Primitive of holomorphic Function $\frac{1}{z}$ on an Annulus.

real-analysis complex-analysis cauchy-integral-formula

Evaluating $\int_0^{\infty} \log(\sin^2(x))\left(1-x\operatorname{arccot}(x)\right) \ dx$

calculus real-analysis integration definite-integrals

Are there sets of zero measure and full Hausdorff dimension?

real-analysis measure-theory

Separation theorem on the space of all complex continuous functions

real-analysis functional-analysis

Let $f,g:X\rightarrow \mathbb{R}$ continuous functions .If $X$ is open set,then the following set is open:$A=\{x \in X;f(x)\neq g(x)\}$

Proving that: $9.9998\lt \frac{\pi^9}{e^8}\lt 10$?

calculus real-analysis approximation approximation-theory transcendental-numbers

How to show that $\lim_{n\to \infty} \frac{a_1 +a_2 + \cdots + a_n}{n} = 0?$ [duplicate]

real-analysis sequences-and-series limits solution-verification alternative-proof