Properties of Continuous Functions

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,

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

Determine the values of $a, b, c$ for which the function defined by

$\begin{array}{ll}f\left(x\right)=\frac{\sin (a+1)x+\sin x}{x},& \text{ for }x<0\\ =c,& \text{ for }x=0\\ =\frac{{\left(x+b{x}^2\right)}^{1/2}-x^{1/2}}{b{x}^{1/2}},& \text{ for }x>0\end{array}$

is continuous at $x=0$.

Find $k$, so that the function $f\left(x\right)$ is continuous at $x=1,$ where

$\begin{array}{ll}f\left(x\right) =k{x}^2,& \text{ for }x\ge 1\\ =4,& \text{ for }x<1\end{array}$

Find the value of $a$ and $b$ such that the function defined by

$\begin{array}{ll}f\left(x\right) =5,& \text{ if }x\le 2\\ =ax+b,& \text{ if }2<x<10\\ =21,& \text{ if }x\ge 10\end{array}$is continuous on at $x=2$ as well as $x=10$.

If $f\left(x\right)$ is defined by $\begin{array}{ll}f\left(x\right)=\sin 2x,& \text{ if }x\le \frac{\pi }{6}\\ =ax+b,& \text{ if }x>\frac{\pi }{6}\end{array}$Find the value of $a$ and $b$, if $f\left(x\right)$ is continuous and differentiable at $x=\frac{\pi }{6}$.

Examine the continuity of the following function at given point

$f\left(x\right)=\begin{array}{ll}x,& \mathrm{for} 0\le x<\frac{1}{2}\\ 1-x,& \mathrm{for} \frac{1}{2}\le x<1\end{array}\} \mathrm{at} x=\frac{1}{2}$

Examine the continuity of the following function at given point

$f\left(x\right)=\begin{array}{ll}\underset{x\to 0}{\lim }\left(\frac{e^{{\displaystyle \frac{1}{x}}}-1}{e^{{\displaystyle \frac{1}{x}}}+1}\right),& \mathrm{for} x\ne 0\\ 1,& \mathrm{for} x=0\end{array}\} \mathrm{at} x=0$

If $\begin{array}{ll}f\left(x\right)=\frac{x^2-9}{x-3}+\alpha ,& \text{ for }x>3\\ =5,& \text{ for }x=3\\ =2x^2+3x+\beta , & \text{ for }x<3\end{array}$ is continuous at $x=3$ then find $\alpha$ and $\beta$.

If $\begin{array}{ll}f\left(x\right)=\frac{\sin \pi x}{x-1}+a,& \text{ for }x\le 1\\ =2\pi ,& \text{ for }x=1\\ =\frac{1+\cos \pi x}{\pi (1-x{)}^2}+b,& \text{ for }x>1\end{array}$ is continuous at $x=1$, find $a$ and $b$.

Examine the continuity of the following function at given point 

$f\left(x\right)=\begin{array}{ll}\frac{x}{\left|x\right|},& \mathrm{for} x\ne 0\\ c,& \mathrm{for} x=0\end{array}\} \mathrm{at} x=0 \left(\mathrm{where} \mathrm{c} \mathrm{is} \mathrm{arbitrary} \mathrm{constant}\right)$

If $\begin{array}{ll}f\left(x\right)=\frac{\sin 4x}{5x}+a,& \text{ for }x>0\\ =x+4-b& \text{ for }x<0\\ =1& \text{ for }x=0\end{array}$ is continuous at $x=0$, find $a$ and $b$.

If $f\left(x\right)=\left\{\begin{array}{l}\begin{array}{ll}{x}^2+\alpha ,& \text{ for }x\ge 0\end{array}\\ \begin{array}{ll}2\sqrt{x^2+1}+\beta ,& \text{ for }x<0\end{array}\end{array}\right.$ and $f\left(\frac{1}{2}\right)=2$, is continuous at $x=0$, find $\alpha$ and $\beta$.

If $f\left(x\right)=\frac{1-\sqrt{3} \tan x}{\pi -6x}$, for $x\ne \frac{\pi }{6}$ is continous at $x=\frac{\pi }{6}$, find $f\left(\frac{\pi }{6}\right)$.

If $f\left(x\right)=\frac{4^x-2^{x+1}+1}{1-\cos x}$, for $x\ne 0$ is continuous at $x=0$, find $f\left(0\right)$.