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?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this communityLet $\Bbb f : R \to R$ 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?
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 $f(0)$.