Continuity and Differentiability

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x) = \begin{cases} x\left(\dfrac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0. \end{cases}$ Then

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=\min \left\{|x|, x^2\right\}$. Then $f$ is

Let $a, b \in \mathbb{R}$ and $f$ be the function defined by $f(x)= \begin{cases}e^x, & \text { if } x \lt 5 \\ a+b x, & \text { if } x \geq 5\end{cases}$ If $f$ is differentiable and $f^{\prime}$ is continuous on…

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=1$ and $|f(x)-f(y)| \leq\left|\sin \left\{(x-y)^2\right\}\right| \text { for all } x, y \in \mathbb{R}$ and let $g$ be the function…

Define $\operatorname{sgn}(x)= \begin{cases}1, & \text { if } x \gt 0, \\ -1, & \text { if } x \lt 0, \\ 0, & \text { if } x=0 .\end{cases}$ Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function defined by…

Suppose $a$ is a real number and $f$ is defined as $f(x)= \begin{cases}|x|^a \sin \left(|x|^{-3}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases}$ Then

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=\min \left(x^3, x^2\right)$, where $\min \left(x^3, x^2\right)$ is the minimum of $x^3$ and $x^2$. Then $f$ is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $|f(x)-f(y)| \leq|x-y|^{\frac{3}{2}} \quad$ for all $x, y \in \mathbb{R}$.

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as follows $f(x)=\left\{\begin{array}{l} e^{-\frac{1}{|x|}}, \quad x \neq 0 \\ 0, \quad x=0 \end{array}\right.$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. For an $\alpha \in \mathbb{R}$, suppose that $\lim _{h \rightarrow 0} \frac{f(\alpha+h)-f(\alpha-h)}{h}=p,$ where $p \in \mathbb{R}$. Then

Let $f: \mathbb{R} \mapsto \mathbb{R}$ be defined by $f(x)=[x]+\sqrt{x-[x]}$. Here, $[x]$ is the greatest integer function for $x \in \mathbb{R}$ and $\sqrt{y}$ is the positive square root of $y$ for $y \in…

Let $f:[-1,1] \rightarrow \mathbb{R}$ be a function such that $|f(x)| \leq x^2$ for all $x \in[-1,1]$. Then