Question

Show that the function $f\left(x\right)=2x-\left|x\right|$ is continuous at $x=0$.

Answer 1

The function $f\left(x\right)=2x-\left|x\right|$ can be written as

$f\left(x\right)=\{\begin{array}{ccc}2x+x& :& x\le 0\\ 2x-x& :& x>0\end{array}$.

Now,

$\underset{x\to 0-}{lim}f\left(x\right)=$$\underset{x\to 0-}{lim}3x=0$.

And,

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

So we've,

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

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