**Lemma** The function $f(x)=\lfloor x \rfloor$ ([[Floor and Ceiling Functions]]) is discontinuous at every point $c \in \mathbb{Z}$. **Proof ([[Discontinuous Function (Epsilon-Delta Definition)]])** Take $\epsilon=1$, $c\in \mathbb{Z}$. Then for $\delta>0$, take $x = c - \frac{1}{2} \delta$ then $|c-x| =\frac{\delta}{2}<\delta$ and $|f(x)-f(c)| = | \lfloor c- \frac{1}{2} \delta \rfloor - c| \geq 1.$ **Proof ([[Discontinuous Function (Sequence Definition)]])** Take $x_{n} = c - \frac{1}{n}$; then $x_{n} \to c$ but $f(x_{n})= c-1$ for all $n$, so $f(x_{n}) \to c-1 \neq c = f(c)$.