**Lemma** The function $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \begin{cases} 0 & x\leq 0 \\ 1 & x> 0 \end{cases}$ $f(x)$ is [[Discontinuous Function|discontinuous]] at $0$. **Proof ([[Discontinuous Function (Epsilon-Delta Definition)]])** Take $\epsilon =1$, then for any $\delta>0$, the point $x = \frac{\delta}{2}$ satisfies $|x-0| = |x| = \frac{\delta}{2} < \delta \quad \text{ and } \quad |f(x)-f(0)|= |f(x)| 1 \geq 1$so $f$ is discontinuous at $x = 0$. **Proof ([[Discontinuous Function (Sequence Definition)]])** Take $x_{n} = \frac{1}{n}$, then $x_{n} \to 0$, but $f(x_{n}) \to 1 \neq 0= f(0)$.