Define Mode function in non continuous region.

Question

Let $\Bbb f : R \to R$ be a function defined as $f(x)=|x|/2x$ $\forall x \in R$ {0}

Can $f(0)$ be defined in a way such that $f$ is continuous at 0?

Answer 1

The function will be continous in 0 $ \Leftrightarrow f(0)=\lim_{x \rightarrow 0^+ }f(x)=\lim_{x \rightarrow 0^- }f(x) $.

In your case, $\lim_{x \rightarrow 0^+ }f(x)=\lim_{x \rightarrow 0^+ }|x|/2x =1/2$ and $\lim_{x \rightarrow 0^- }f(x)=\lim_{x \rightarrow 0^- }|x|/2x=-1/2$.

So the function $f(x)$ can't be continuous, regardless the value of $f(0)$.