## Continuity at a Point

Suppose a function is defined by\[f(x)= \left\{ \begin{array}{c} x \; x \lt 0 \\ 1 \; x=0 \\ x \; x \gt 0 \end{array} \right. \]

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 \[x=0\]

.