The step function $f: \mathbb{R}\to \mathbb{R}$
$f(x)=\begin{cases}0 & x < 0 \\ 1 & x \geq 0 \end{cases}$ The constant function is continuous so $f(x)$ is continuous for all $x<0$ and $x\geq0$ But on the other hand when x =0 for any $\delta>0$ and any $x\in(-\delta,0)$ $|f(x)-f(0)|=1$ and once we take $\epsilon \leq1$ the function is not continuous.But f is constant at $0$. I must be doing mistake but can anyone help me with organizing my knowledge about continuity?