Points of Discontinuity

Examine the continuity or discontinuity of the following functions

(i) $f\left(x\right)=\left[x\right]+\left[-x\right]$             (ii) $g\left(x\right)=\underset{n\to \infty }{\lim }\frac{x^{2n}-1}{x^{2n}+1}$

Here $\left[·\right]$, denotes the greatest integer function.

Find the set of points, where, $f\left(x\right)=\tan 2x$, is discontinuous

If $\alpha ,\beta \left(\alpha <\beta \right)$ are the points of discontinuity of the function $f\left(f\left(f\left(x\right)\right)\right),$ where $f\left(x\right)=\frac{1}{1-x}$, then the set of values of $a$ for which the points $\left(a,a^2\right)$ lie on the same side of the line $x+2y-3=0$, is

$f\left(x\right)=\left\{\begin{array}{l}\begin{array}{ccc}-1& ,& x<0\\ 0& ,& x=0\end{array}\\ \begin{array}{ccc}1& ,& x>0\end{array}\end{array}\right.$ and $g\left(x\right)=x(1-x^2),$ then $f\left(g\right(x\left)\right)$ is continuous for

Which of the following function(s) has/have the same range?

$f$ is a continuous function in $\left[a, b\right], g$ is a continuous function in $\left[b, c\right]$. A function $h\left(x\right)$ is defined as $h\left(x\right)=\left\{\begin{array}{c}f\left(x\right) \mathrm{for} x\in \left[a, b\right)\\ g\left(x\right) \mathrm{for} x\in (b, c]\end{array}\right.$. If $f\left(b\right)=g\left(b\right)$, then:

The function $f\left(x\right)=\left[\left|x\right|\right]-\left|\left[x\right]\right|$, where $\left[x\right]$ denotes greatest integer function

Find the number of point(s) of discontinuity of $f\left(x\right)=\left[x^{\frac{1}{x}}\right], x>0$, where $\left[.\right]$ denotes the greatest integral function.

Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{e^{x^2}},& \mathrm{i}\mathrm{f} x\ne 0\\ 0,& \mathrm{i}\mathrm{f} x=0\end{array}\right.$
$f^{\text{'}}\left(0\right)$ is equal to:

The function $f\left(x\right)=\left(x^2-1\right)\left|x^2-6x+5\right|+\mathrm{c}\mathrm{o}\mathrm{s}\left|x\right|$ is not differentiable at

The function $f\left(x\right)={\left[x\right]}^2-\left[x^2\right]$ (where $\left[y\right]$ is the greatest integer less than or equal to $y$) is discontinuous at:

If the functions $f: R\to R$ and $g: R\to R$ are such that $f\left(x\right)$ is continuous at $x=\alpha$, and $f\left(\alpha \right)=a$ and $g\left(x\right)$ is discontinuous at $x=a$ but $g\left(f\left(x\right)\right)$ is continuous at $x=\alpha$, then $(f\left(x\right)$ and $g\left(x\right)$ are non-constant functions.)