The points of discontinuity of $\tan x$ are

Important Questions on Continuity and Differentiability

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Let $f :\left[0, \pi \right]\to R$ be defined as

$f\left(x\right)=\left\{\begin{array}{ccc}\sin x,& \text{if}& x \text{is irrational and} x\in \left[0,\pi \right]\\ {\tan }^{2}x,& \text{if}& x\text{ is rational and} x\in \left[0,\pi \right]\end{array}\right.$,

The number of points in $\left[0, \pi \right]$ at which the function $f$ is continuous is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If $f\left(x\right)=\left\{\begin{array}{ccc}\log {\left({\sec }^{2}x\right)}^{{\cot }^{2}x}& \mathrm{f}\mathrm{o}\mathrm{r}& x\ne 0\\ K& \mathrm{f}\mathrm{o}\mathrm{r}& x=0\end{array}\right.$ is continuous at $x=0$ then $K$ is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

The value of $k$ which the function $f\left(x\right)= \left\{\begin{array}{cc}{\left(\frac{4}{5}\right)}^{\frac{\tan 4x}{\tan 5x}},& 0<x<\frac{\pi }{2}\\ k+\frac{2}{5},& x=\frac{\pi }{2}\end{array}\right.$ is continuous at $x=\frac{\pi }{2},$ is

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Let $a,b\in R,\left(a\ne 0\right).$ If the function $f$, defined as

$f\left(x\right)=\left\{\begin{array}{c}\frac{2x^2}{a}, 0\le x<1\\ a, 1\le x<\sqrt{2}\\ \frac{2b^2-4b}{x^3}, \sqrt{2}\le x<8\end{array}\right.$,is continuous in the interval $\left[0, \infty \right)$, then an ordered pair $\left(a,b\right)$ can be 

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Let $k$ be a non - zero real number. If $f\left(x\right)=\left\{\begin{array}{l}\frac{{(e^x-1)}^2}{\sin \hspace{0.17em}\left(\frac{x}{k}\right)\log \hspace{0.17em}\left(1+\frac{x}{4}\right)}\begin{array}{cc},& x\hspace{0.17em}\ne \hspace{0.17em}0\end{array}\\ \begin{array}{cc}12\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em},& x\hspace{0.17em}=\hspace{0.17em}0\end{array}\end{array}\right.\hspace{0.17em}$ is a continuous function at $x=0$, then the value of $k$ is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If $f\left(x\right) =\left\{\begin{array}{ll}\frac{1}{\left|x\right|};& \left|x\right| \ge 1\\ a{x}^2+b;& –1<x<1\end{array}\right.$ is differentiable $\forall x$ , then one of the value of $a$ and $b$ is-

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If $f\left(x\right) = \left\{\begin{array}{ccc}x& \text{for}& x\le 0\\ 0& \text{for}& x>0\end{array}\right.$ then $f\left(x\right)$ at $x=0$ is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Consider the function $f\left(x\right)=\left\{\begin{array}{ccc}\frac{x+5}{x-2}& \mathrm{i}\mathrm{f}& x\ne 2\\ 1& \mathrm{i}\mathrm{f}& x=2\end{array}\right.$ . Then $f\left(f\left(x\right)\right)$ is discontinuous

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If the function $f$ defined as $f\left(x\right)=\frac{1}{x}-\frac{k-1}{e^{2x}-1}, x\ne 0$ is continuous at $x=0$, then ordered pair $\left(k,f\left(0\right)\right)$ is equal to

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Let $f:\left[-\mathrm{1,3}\right]\to \mathrm{R}\mathrm{ }$ be defined as
$f\left(x\right)=\left\{\begin{array}{c}\left|x\right|+\left[x\right],\\ x+\left|x\right|,\\ x+\left[x\right],\end{array}\begin{array}{c}-1\le x<1\\ 1\le x<2\\ 2\le x\le 3,\end{array}\right.$
Where $\left[\mathrm{t}\right]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If $f\left(x\right)=1-x, \mathrm{for} 0<x\le 1=k$, for $x=0$ is continuous at $x=0$, then $k=$_____

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If the function $f$ defined on $\left(\frac{\pi }{6},\frac{\pi }{3}\right)$ by $f\left(x\right)=\left\{\begin{array}{c}\frac{\sqrt{2}\mathrm{c}\mathrm{o}\mathrm{s}x-1}{\mathrm{c}\mathrm{o}\mathrm{t}x-1}, x\ne \frac{\pi }{4}\\ k, x=\frac{\pi }{4}\end{array}\right.$ is continuous, then $k$ is equal to

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If the function $f\left(x\right)=\left\{\begin{array}{cc}\frac{\sqrt{2+cosx}-1}{{\left(\pi -x\right)}^2},& x\ne \pi \\ k,& x=\pi \end{array}\right.$ is continuous at $x=\pi$, then $k$ equals

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If $f\left(x\right)=\left[x\right]-\left[\frac{x}{4}\right],x\in R,$ where $\left[x\right]$ denotes the greatest integer function, then:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If the function $f\left(x\right)=\left\{\begin{array}{cc}\frac{\left(e^{kx}-1\right)\tan kx}{4x^2},& x\ne 0\\ 16,& x=0\end{array}\right.$ is continuous at $x=0$, then $k$ = ….

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If $f\left(x\right)=\left\{\begin{array}{ccc}\frac{\mathrm{s}\mathrm{i}\mathrm{n}(p+1)x+\mathrm{s}\mathrm{i}\mathrm{n}x}{x}& ,& x<0\\ q& ,& x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}}& ,& x>0\end{array}\right.$ is continuous at $x=0$ , then the ordered pair $(p, q)$ is equal to:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If the function $f\left(x\right)=\left\{\begin{array}{ccc}{\left[\tan \left(\frac{\pi }{4}+x\right)\right]}^{\frac{1}{x}}& \mathrm{f}\mathrm{o}\mathrm{r}& x\ne 0\\ k& \mathrm{f}\mathrm{o}\mathrm{r}& x=0\end{array}\right.$ is continuous at $x=0,$ then $k=?$

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be defined by $f\left(x\right)=x-\left[x\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x.$ Also let $S$ denote the range of $f.$ Which of the following is TRUE?

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Let $f:R\to R$ be a continuous function. Then, $f$ is surjective if