Continuity at a Point
\[f(x)= \left\{ \begin{array}{c} x \; x \lt 0 \\ 1 \; x=0 \\ x \; x \gt 0 \end{array} \right. \]
We can define the limit of a function
\[f(x)\]
at a point \[x_0\]
in two ways - as \[x\]
tends to \[x_0\]
from below, written \[x \rightarrow x_0^{{}-{}}\]
and as \[x\]
tends to \[x_0\]
from above, written \[x \rightarrow x_0^{{}+{}}\]
. \[f(x)\]
is continuous at \[x_0\]
if and only if \[lim_{x \rightarrow x_0^{{}-{}}} f(x)= lim_{x \rightarrow x_0^{{}+{}}} f(x)\]
.As
\[x \rightarrow 0^{{}-{}}\]
, \[f(x) =x \rightarrow 0\]
and as \[x \rightarrow 0^{{}+{}}\]
, \[f(x) =x \rightarrow 0\]
so \[f(x)\]
is continuous at \[x=0\]
. 
