Example

a) $f: \mathbb{R} \to \mathbb{R}$
$x \mapsto 4x - 1$ is continuous.
Let $(x_n)_{n \in \mathbb{N}}$ be a sequence with $\lim_{n \to \infty} x_n = x_0$
$f(x_n) = 4x_n - 1$
$\lim_{n \to \infty} (4x_n - 1) = 4 \cdot \lim_{n \to \infty} x_n - 1 = 4x_0 - 1 = f(x_0)$

b) $f: \mathbb{R} \to \mathbb{R}$

$$x \mapsto \left\{ \begin{array}{cl} x & \text{if } x < 1 \\ x-2 & \text{if } x \geq 1 \end{array}\right.$$

If a function $f$ is continuous in $x_0$ for all $x_0 \in U$, then we say that $f$ is continuous on $U$.

If $f$ is not continuous in $x_0$, $f$ is discontinuous in $x_0$.