Question

Check the continuity of the function function $f\left(x\right)=\left|x\right|$ at $x=0$.

Answer 1

Given the function $f\left(x\right)$ can be written as,

$f\left(x\right)=\{\begin{array}{c}x;\text{if}x\ge 0\\ -x;\text{if}x<0\end{array}$.

Still,

$\underset{x\to 1+}{lim}f\left(x\right)=0=\underset{x\to 1-}{lim}f\left(x\right)=f\left(0\right)$.

So the function $f\left(x\right)$ is continuous at $x=0$.