Consider the function defined as follows for $x\ne 0$. In each case what choice of $f\left(0\right)$ will make the function continuous at $x=0$?

$f\left(x\right)=\frac{\sqrt[3]{1+x}-1}{x}$

Examine the following function $f\left(x\right)=\left\{\begin{array}{l}x \sin \frac{1}{x} , x\ne 0\\ 0 , x=0\end{array}\right.$for continuity at $x=0$.

Show that the function $f\left(x\right)=\left|x\right|+\left|x-1\right|$, $x\in R$, is continuous both at   $x=0$ and $x=1$.

Examine the function for continuity:

$f\left(x\right)=\left\{\begin{array}{l}x , \mathrm{if} x\ge 0\\ x^2 , \mathrm{if} x<0.\end{array}\right.$

Examine the function for continuity:

$f\left(x\right)=\left\{\begin{array}{l}x+5, \mathrm{if} x\le 1\\ x-5, \mathrm{if} x>1\end{array}\right.$

Examine the function for continuity:

$f\left(x\right)=\left\{\begin{array}{l}2x-1, \mathrm{if} x<2\\ \frac{3}{2}x , \mathrm{if} x\ge 2\end{array}\right.$