Continuity and Differentiability

Let $f:\left[0,\infty\,\right)\rightarrow\,\left[0,3\right]$ be a function defined by$f\left(x\right)=\left{\begin{matrix} \max\,\left{\sin\,t:0\leq\,t\leq\,\pi\,\right},x\in\,\left[0,\pi\,\right] \\ 2+\cos\,x,x>\pi\,…

Let $\mathrm{f}(x)=\left\{\begin{array}{lc}3 x, & x \lt 0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x\gt2,\end{array}\right.$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the…

Let$a\in\,\mathbb{Z}$and$\left[t\right]$be the greatest integer$\leq\,t$, then the number of points, where the function$f\left(x\right)=\left[a+13\sin\,x\right],x\in\,\left(0,\pi\,\right)$is not differentiable,…

Let$f\left(x\right)=\left|2x^{2}+5\left|x\right|-3\right|,x\in\,R$.If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to:

Let$f$and$g$be two functions defined by$f\left(x\right)=\left{\begin{matrix} x+1,x<0 \\ \left|x-1\right|,x\geq\,0 \end{matrix}\right$and$g\left(x\right)=\left{\begin{matrix} x+1,x<0 \\ 1,x\geq\,0…

The number of points where the function$f\left(x\right)=\left{\begin{matrix} \left|2x^{2}-3x-7\right| & \mathrm{if}x⩽-1 \\ \left[4x^{2}-1\right] & \mathrm{if}-1<x<1 \\ \left|x+1\right|+\left|x-2\right| & \mathrm{if}x⩾1…

The function $f\left(x\right)=\left{\begin{matrix} \frac{\pi\,}{4}+\tan\,^{-1}x, & \left|x\right|\leq\,1 \\ \frac{1}{2}\left(\left|x\right|-1\right), & \left|x\right|>1 \end{matrix}\right$ is :

Let $a,b\in\,R,\left(a\neq\,0\right).$ If the function $f$, defined as $f\left(x\right)=\left{\begin{matrix} \frac{2x^{2}}{a},0\leq\,x<1 \\ a,1\leq\,x<\sqrt{2} \\ \frac{2b^{2}-4b}{x^{3}},\sqrt{2}\leq\,x<8…

Let$f:\left[0,3\right]\rightarrow\,R$be defined by$f\left(x\right)=\min\,\left{x-\left[x\right],1+\left[x\right]-x\right}$where $\left[x\right]$ is the greatest integer less than or equal to $x.$Let $P$ denote the set…

Let the function, $f(x)= \begin{cases}-3 a x^2-2, & x \lt 1 \\ a^2+b x, & x \geqslant 1\end{cases}$ be differentiable for all $x \in \mathbf{R}$, where $\mathbf{a}\gt1, \mathbf{b} \in \mathbf{R}$. If the area of the…

Let $f:\left(-1,1\right)\rightarrow\,R$ be a function defined by $f\left(x\right)=max\left{-\left|x\right|,-\sqrt{1-x^{2}}\right}.$ If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly

Let $f\left(x\right)=\left{\begin{matrix} max\left(\left|x\right|,x^{2}\right), & \left|x\right|\leq\,2 \\ 8-2\left|x\right|, & 2<\left|x\right|\leq\,4 \end{matrix}.\right$Let $S$ be the set of points in the interval…