Recent questions and answers in Analysis

Prove or disprove each of the statements below.Let $f: \mathbb{R}^{2} \longrightarrow \mathbb{R}$ be a continuous function that takes both positive and negative values. T...

Let $f:[0,1] \rightarrow \mathbb{R}$ be such that\[f(x)=\left\{\begin{array}{ll}\frac{1}{2^{n}}, & \text { if } x=\frac{p}{2^{n}}, \text { where } p \text { is an odd int...

Let $x_{n}=\left(1-\frac{1}{\sqrt{n}}\right)^{n} \exp \left(n^{\frac{1}{4}}\right)$. Which of the following statements is correct?$\displaystyle \lim _{n \rightarrow \inf...

What is the number of functions $f: \mathbb{R} \backslash\{0,1\} \rightarrow \mathbb{R} \backslash\{0,1\}$ such that $|f(x)-f(y)|=|x-y|$ for all $x, y \in \mathbb{R} \bac...