Question

Show that f(x) = 2x − |x| is continuous at x = 0

Answer 1

The given function can be rewritten as:
$f\left(x\right)=\left\{\begin{array}{c}2x-x,\mathrm{when}x>0\\ 2x+x,\mathrm{when}x<0\\ 0,\mathrm{when}x=0\end{array}\right.$
​$\Rightarrow$ $f\left(x\right)=\left\{\begin{array}{c}x,\mathrm{when}x>0\\ 3x,\mathrm{when}x<0\\ 0,\mathrm{when}x=0\end{array}\right.$

We observe
(LHL at x = 0) = $\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0-h\right)=\underset{h\to 0}{\lim }f\left(-h\right)$$=\underset{h\to 0}{\lim }\left(-3h\right)=0$


(RHL at x = 0) = $\underset{x\to 0^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0+h\right)=\underset{h\to 0}{\lim }f\left(h\right)$= $\underset{h\to 0}{\lim }\left(h\right)=0$

And, $f\left(0\right)=0$


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

Thus, f(x) is continuous at x = 0.