Define Mode function in non continuous region. [closed]

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?

• Actually i was trying to draw graph of different curves and i found out this question.. in every way its a discontinuous function that i found where values would be 1/2 when x>0 and -1/2 when x<0 . but at x=0 its in my opinion its undefined.. but the question has asked otherwise. so after every try i found it unable to define any way.. Apr 18, 2019 at 12:20

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 $$f(0)$$.