Question

Examine the continuity of function $f\left(x\right)=|x-2|$ at $x=2$.

Answer 1

$\Rightarrow {lim}_{x\to 2^{-}}f\left(x\right)=\underset{h\to 0}{lim}f(2-h)=\underset{h\to 0}{lim}|2-h-2|=\underset{h\to 0}{lim}|-h|=0$

$\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(2+h)=\underset{x\to 0}{lim}|2+h-2|=|h|=0$

and $f\left(2\right)=|2-2|=0$

$\Rightarrow f$ is continuous at $x=2$.