Properties of Continuous Function

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 $R$

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

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,

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

Show that the function defined by $f\left(x\right)=\left|\cos x\right|$  is continuous function 

Prove that composition of two continuous function is continuous.

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

Let $f\left(x\right)=\{\begin{array}{cc}x+a,& \mathrm{if }x<0\\ \left|x-1\right|,& \mathrm{if }x\ge 0\end{array}$

and $g\left(x\right)=\{\begin{array}{cc}x+1,& \mathrm{if }x<0\\ {\left(x-1\right)}^2+b,& \mathrm{if }x\ge 0\end{array}$

where $a$ and $b$ are non-negative real numbers. If the composite function $gof\left(x\right)$ is continuous for all real $x,$ then the values of $a+b$ 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 :

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

Let $f\left(x\right)= \left\{\begin{array}{cc}\frac{1-\sin \pi x}{1+\cos 2\pi x},& x<\frac{1}{2}\\ p,& x=\frac{1}{2}\\ \frac{\sqrt{2x-1}}{\sqrt{4+\sqrt{2x-1}-2}},& x>\frac{1}{2}\end{array}\right.$. Then value of $p$ for which the function is continuous at $x=\frac{1}{2}$ is

If $g\left(x\right)={\left(4{\cos }^{4}x-2\cos 2x-\frac{1}{2}\cos 4x-x^7\right)}^{\frac{1}{7}}$ then the value of $g\left(g\left(100\right)\right)$ is equal to

Let $f\left(x\right)=\frac{x^3}{4}-\sin \pi x+3.$ If $f\left(x\right)$ takes the value $\alpha$ on $\left[-2, 2\right],$ then $\alpha$ is equal to

If $f\left(x\right)=\frac{1}{1-x}$ , then the points of discontinuity of the function $f^n\left(x\right)$ is/are, where $f^n=fof\dots \dots of \left(n \mathrm{times}\right)$, are