# Function Discontinuities A **function is said to be discontinuous at $x$ if it is not defined for $x$, or if its graph jumps at that point**. It is then said that it has a discontinuity at $x$. **Three types of discontinuities** exist: 1. **Point discontinuity**. When the function is not defined for a certain $x$ e.g. $f(x) = \frac {(x-1)x} {(x-1)}$ is not defined at $x=1$, but it otherwise matches $x$. 2. **Jump discontinuity**. When the function's graph suddenly jumps e.g. $f(x) =\begin{cases} x \le 2 & x^2 \\ x > 2 & \sqrt x \end{cases}$ 3. **Asymptote**. When approaching a value leads to division by 0 e.g. $f(x) = \frac 1 x$. A function is said to be continuous (on its entire domain) if it does not have any discontinuities. Put differently **a function is continuous if for all $a$ elements of its domain the following holds $\lim_{x \to a} f(x) = f(a)$.** Two valuable theorems govern continuous functions: [[The Intermediate Value Theorem]] and [[The Extreme Value Theorem]].