Properties of Continuous Functions

Let $f: R\to R$ be any function and $g: R\to R$ is defined by $g\left(x\right)=\left|f\left(x\right)\right|$ for all $x$, then $g$ is

Find the constants $a$ and $b$ so that the function $f$ defined below is continuous in $R$

 $f\left(x\right)=\left\{\begin{array}{ll}1,& x⩽3\\ ax+b,& 3<x<5\\ 7,& x⩾5\end{array}\right.$

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?

A function $f\left(x\right)$ is continuous over a closed interval $x\in \left[1, 4\right]$.

What can you conclude using the extreme value theorem about a function that is continuous over the closed interval $x\in \left[1, 4\right]$?

A function has a maximum and a minimum in the closed interval $\left[a, b\right]$; therefore, the function is continuous in $\left[a, b\right]$.

The converse of extreme value theorem is always true.

A function is continuous over the interval $\left[a, b\right]$; therefore, the function has a maximum and a minimum in the closed interval.

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

If the function $f\left(x\right)$ is continuous on its domain $[-2,2]$ when,

$f\left(x\right)=\left\{\begin{array}{l}\frac{\sin ax}{x}+2,\text{ for }-2\le x<0\\ \begin{array}{c}3x+5,\text{ for }0\le x\le 1\\ \sqrt{x^2+8}-b,\text{ for }1<x\le 2\end{array}\end{array}\right.$

If $f\left(x\right)=\left\{\begin{array}{l}\frac{\log \left(1+mx\right)-\log \left(1-nx\right)}{x}; x\ne 0\\ \lambda ; x=0\end{array}\right.$, is continuous at $x=0,$ the value of $\lambda$ will be

For a real number $y$, let $\left[y\right]$ denotes the greatest integer less than or equal to $y$. Let $f\left(x\right)=\frac{\tan \left(\pi \left[x-\pi \right]\right)}{1+[x{]}^2}$. Then

If the function $f$ defined by $f\left(x\right) =\left\{\begin{array}{cc}{\left(\cos x\right)}^{\frac{1}{x}},& x\ne 0\\ k,& x=0\end{array}\right.$ is continuous at $x=0$, then the value of $k$ is

Identify which of the following is correct for the function $f\left(g\left(x\right)\right)$, if $f\left(x\right)=\left|\sin x\right|$ and $g\left(x\right)=x^3$.

The function $T\left(x\right)=f\left(g\left(f\left(x\right)\right)\right)+g\left(f\left(g\left(x\right)\right)\right)$ is :

The function $f\left(x\right)=\frac{x^3}{8}-\sin \pi x+4$ in $[-4,4]$ does not take the value

The number of points of discontinuity of $f\left(x\right)=[2x{]}^2-{\left\{2x\right\}}^2$ (where $\left[.\right]$ denotes the greatest integer function and $\left\{\right\}$ is fractional part of $x$) in the interval $(-2, 2)$ is

Number of points of discontinuity of $f\left(x\right)=\left[{\sin }^{-1}x\right]-\left[x\right]$ in its domain is equal to (where $\left[.\right]$ denotes the greatest integer function)

If $f\left(x\right)=\left[x\right]{\left(\sin kx\right)}^p$ is continous for real $x$, then (where $\left[.\right]$ represents the greatest integer function)

If $f\left(x\right)=\frac{x}{2}-1,$ then on the interval $\left[0,\pi \right]$ which one of the following is correct?

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