Show that the function `f(x)=2x-|x|` is continuous at `x=0.`

Answer

The function `f(x)` is continuous at `x=0`.

Explanation

The function `f(x)=2 x-|x| ` can be re-written as

f(x)= \begin{cases}2 x+x & : x \leq 0 \\ 2 x-x & : x>0\end{cases}
$$ Now, $$
\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} 3 x=0 .
$$
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