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

Important Questions on Continuity and Differentiability

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 5 September 2020 Shift 1>Q 31

If the function $f\left(x\right)=\left\{\begin{array}{cl}{k}_1(x-\pi {)}^2-1,& x\le \pi \\ k_2\cos x,& x>\pi \end{array}\right.$ is twice differentiable, then the ordered pair $\left(k_1,k_2\right)$ is equal to:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 5 September 2020 Shift 1>Q 32

Let $f\left(x\right)=x·\left[\frac{x}{2}\right],$ for $-10<x<10,$ where $\left[t\right]$ denotes the greatest integer function. Then the number of points of discontinuity of $f\left(x\right)$ is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 6 September 2020 Shift 1>Q 33

 If $f(x+y)=f\left(x\right) f\left(y\right)$ and $\underset{x=1}{\overset{\infty }{\Sigma }}f\left(x\right)=2, x,y\in N$, where $N$ is the set of all natural numbers, then the value of $\frac{f\left(4\right)}{f\left(2\right)}$ is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 6 September 2020 Shift 1>Q 34

Let $f:R\to R$ be defined as $f\left(x\right)=\left\{\begin{array}{ccc}{x}^5\sin \left(\frac{1}{x}\right)+5x^2& ,& x<0\\ 0& ,& x=0\\ x^5\cos \left(\frac{1}{x}\right)+\lambda x^2& ,& x>0\end{array}\right.$. The value of $\lambda$ for which $f^{"}\left(0\right)$ exists, is___.
 

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 6 September 2020 Shift 2>Q 35

For all twice differentiable functions $\mathrm{f} :\mathbf{ }\mathbf{R}\mathbf{\to }\mathbf{R}$, with $\mathrm{f}\left(0\right)=\mathrm{f}\left(1\right)=\mathrm{f}\text{'}\left(0\right)=0$,

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 6 September 2020 Shift 2>Q 36

Let $f:R\to R$ be a function defined by $f\left(x\right)=\max \left\{x,x^2\right\}$. Let $S$ denotes the set of all points in $R$, where $f$ is not differentiable. Then

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 6 September 2020 Shift 2>Q 37

Suppose that a function $\mathrm{f}:\mathbf{R}\mathbf{\to }\mathbf{R}$ satisfies $\mathrm{f}\left(\mathrm{x}+\mathrm{y}\right)=\mathrm{f}\left(\mathrm{x}\right) \mathrm{f}\left(\mathrm{y}\right)$ for all $\mathrm{x}, \mathrm{y} \mathrm{\epsilon } \mathrm{R}$ and $\mathrm{f}\left(1\right)=3$. If $\sum _{\mathrm{i}=1}^{\mathrm{n}}\mathrm{f}\left(\mathrm{i}\right)=363$, then $\mathrm{n}$ is equal to ..... .

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 7th January 2020 Shift 1>Q 38

Let the function $,\mathrm{}f:\left[-7,0\right]\to R$ be continuous on $\left[-7,0\right]$ and differentiable on $\left(-7,0\right).$ If $f\left(-7\right)=-3$ and $f^{\text{'}}\left(x\right)\le 2$ for all $x\in \left(-7,0\right),$ then for all such functions $f, f\left(-1\right)+f\left(0\right)$ lies in the interval