one-sided continuity of a funtion

The definition of the continuity of a function is as follow.

The function $f$  is continuous, if for any $\epsilon>0$  there is a $\delta>0$  such that
\[ |f(x)-f(c)|<\epsilon  \]
for every points $x$  for which $|x-c|<\delta$ .

It means that $\lim_{x\rightarrow -c}f(x)=f(c)$  and $\lim_{x\rightarrow +c}f(x)=f(c)$ .
By the preceding definition of one-sided limits, we are able to expand the continuity or discontinuity of a function.

A function $f(x)$  is continuous from the right at point $c$  if $\lim_{x\rightarrow +c}f(x)=f(c)$ .
Similarly, A function $f(x)$  is continuous from the left at point $c$  if $\lim_{x\rightarrow -c}f(x)=f(c)$ .

There are three kinds of discontinuity at point $c$ .

(1)Removable discontinuity :  $\lim_{x\rightarrow c}f(x)$  exists. But $\lim_{x\rightarrow c}f(x)\ne f(c)$ . If we can redefine the function $f$ except a point $c$ , the discontinuity will be removed.
\[  (\mbox{example})\quad f(x)=\frac{x^2+x-2}{x-1} \]

(2)Jump or Step discontinuity : One-sided limits exists.
\[ (\mbox{example})\quad f(x)=\left\{ \begin{array}{cccc}
 -x^2 & (x<0)& & \\
 x^2+1 & (x\geq 0)& &
\end{array}
\right. \]
This example means $f(x)$ is continuous from the right at point 0, but not from the left.

(3)Infinite or essential discontinuity : One or both of the one-sided limits do not exist or infinite.
\[  (\mbox{example})\quad f(x)=\frac{1}{x-1} \]