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?
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1$\begingroup$ 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.. $\endgroup$– Debayan BairagiApr 18, 2019 at 12:20
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1 Answer
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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 $f(0)$.
