Amit M Agarwal Solutions for Chapter: Continuity and Differentiability, Exercise 5: Proficiency in 'Continuity and Differentiability' Exercise 1

Let $\left[x\right]$ denotes the greatest integer less than or equal to $x$. If $f\left(x\right)=\left[x\mathrm{s}\mathrm{i}\mathrm{n\pi }x\right]$, then $f\left(x\right)$ is:

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

The function $f\left(x\right)=\sqrt{1-\sqrt{1-x^2}}$

Consider the function $f\left(x\right)=\left|x^3+1\right|$. Then,

$f$ 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:

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

If $f\left(x\right)=\mathrm{s}\mathrm{e}\mathrm{c}2x+\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}2x$, then $f\left(x\right)$ is discontinuous at all point in:

Let $f\left(x\right)=\left\{\begin{array}{cc}{x}^n\sin \left(\frac{1}{x^2}\right)& ,x\ne 0\\ 0& ,x=0\end{array},\left(n\in I\right)\right.$, then