Continuity and Differentiability

Let $\(f(x)=|\sin$ x|\). Then

Let $f:R\rightarrow\,R$ be a function defined as $f\left(x\right)=\left{\begin{matrix} 5,ifx\leq\,1 \\ a+bx,if1<x<3 \\ b+5x,if3\leq\,x<5 \\ 30,ifx\geq\,5 \end{matrix}\right$ Then $f$ is:

If the function $f\left(x\right)=\left{\begin{matrix} a\left|\pi\,-x\right|+1,x\leq\,5 \\ b\left|x-\pi\,\right|+3,x>5 \end{matrix}\right$ is continuous at $x=5,$ then the value of $a-b$ is:

Let $f:R\rightarrow\,R$ satisfy the equation $f(x+y)=f(x)\cdot\,f(y)$ for all $x,y\in\,R$ and $f(x)\neq\,0$for any $x\in\,R$. If the function $f$ isdifferentiable at $x=0$and $f^{'}(0)=3$,…

If the function $f\left(x\right)=\left{\begin{matrix} \frac{\sqrt{2+\operatorname{cos}\,x}-1}{\left(\pi\,-x\right)^{2}}, & x\neq\,\pi\, \\ k, & x=\pi\, \end{matrix}\right$is continuous at $x=\pi\,$,then $k$ equals

Let $m$ and $n$ be the number of points at which the function $f(\mathrm{x})=\max \left\{\mathrm{x}, \mathrm{x}^3, \mathrm{x}^5, \ldots ., \mathrm{x}^{21}\right\}, \mathrm{x} \in \mathbb{R}$, is not differentiable and…

Consider the function $f(x)=\left\{\begin{array}{l} \frac{\sin 2 x}{5 x} \text { if } x \neq 0 \\ \frac{2}{15} \text { if } x=0 \end{array}\right.$ Which one of the following is correct in respect of the function?

For what value of $k$ is the function $f(x)=\left\{\begin{array}{ccc} 2 x+\frac{1}{4} & , & x \lt 0 \\ k & , & x=0 \text { continuous? } \\ \left(x+\frac{1}{2}\right)^2 & , & x \gt 0 \end{array}\right.$

If $f(x)$ is differentiable at $x=a$, then consider the following statements : I. $f(x)$ is continuous at $x=a$ II. $\lim\limits_{x\to a}f(x)=f(a)$ Which of the statements given above is/are correct ?

Paragraph: Let $f(x)=[x]$, where $[$.$] is the greatest integer function and $$g(x)=\sin$ x$ be two real valued functions over $R$. Question: Which of the following statements is correct?$

If the function $f: R \rightarrow R$ defined by $f(x)=\left\{\begin{array}{cc}\frac{\sin (a+1) x+\sin x}{x}, & x < 0 \\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3 / 2}}, & x=0 \text { is }\end{array}\right.$ continuous on $[R$,…

$\(f(x)=\left|\log$ $_e\right|$ x \|\) is differentiable at