Real function/Indicator function not negative/Not continuous/Example

We consider the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,$

given by

${}f(x)={\begin{cases}0,{\text{ if }}x<0\,,\\1,{\text{ if }}x\geq 0\,.\end{cases}}\,$

This function is not continuous in ${}0$. For ${}\epsilon ={\frac {1}{2}}$ and every positive ${}\delta$, there exists a negative number ${}x'$ such that ${}d(0,x')=\vert {x'}\vert \leq \delta$. But for such a number we have ${}d(f(0),f(x'))=d(1,0)=1\not \leq {\frac {1}{2}}$.