Continuity and Differentiability

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

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

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 $[R$,…

If $\alpha \in \mathbb{R}-\{-1\}$ and $\mathrm{f}(\mathrm{x})=|(|\mathrm{x}|+\alpha)(|\mathrm{x}|-1)|$, then the number of points at which $f(x)$ is not differentiable, is

If $f\left(x\right)=\operatorname{Max}\,\left{3-x,3+x,6\right}$ is not differentiable at $x=a$, and $x=b$, then $\left|a\right|+\left|b\right|=$

Assertion (A) $f(x)=|x|$ is differentiable at $x=a \neq 0$ and continuous but not differentiable at $x=0$ Reason (R) If a function is differentiable at a point, then it is continuous at the point. But converse is not…

If a real valued function $f(x)=\left\{\begin{array}{cc}(1+\sin x)^{\operatorname{cosec} x}, & -\pi / 2 < x < 0 \\ a, & x=0 \\ \frac{e^{2 / x}+e^{3 / x}}{a e^{2 / x}+b e^{3 / x}}, & 0 < x < \pi / 2\end{array}\right.$,…

If the function $f$ defined by $f(x)=\left\{\begin{array}{cc} \frac{1-\cos 4 x}{x^2}, & x 0 \end{array}\right.$ is continuous at $\mathrm{x}=0$, then $\mathrm{a}=$

If $f(x)=\left\{\begin{array}{cll}\frac{\left(e^{a x}-1\right) \log (1+x)}{\sin ^2 x} &, & \text { if } x>0 \\ 2 &, & \text { if } x=0 \\ \frac{\cos 4 x-\cos b x}{\tan ^2 x} &, & \text { if } x < 0\end{array}\right.$ is…

If a real valued function $f(x)=\left\{\begin{array}{cll}e^{\frac{\sin a(x-[x])}{x-[x]}}, & \text { if } & x 1\end{array}\right.$ is continuous at $\mathrm{x}=1$, then $\mathrm{b} \sin \mathrm{a}=$ ([x] denotes the…

If a real valued function $f(x)= \begin{cases}\log (1+[x]), & x \geq 0 \\ \sin ^{-1}[x], & -1 \leq x < 0 \\ k([x]+|x|), & x < -1\end{cases}$ is continuous at $\mathrm{x}=-1$, then $\mathrm{k}=$

If $f:(-7,7) \rightarrow \mathbb{R}$ is defined by $f(x)=[x]$ for all $x \in(-7,7)$, then the number of discontinuities of $f$ is