# What is a continuous function?

##### 1 Answer
Sep 13, 2015

A continuous function is a function that is continuous at every point in its domain.

That is $f : A \to B$ is continuous if $\forall a \in A , {\lim}_{x \to a} f \left(x\right) = f \left(a\right)$

#### Explanation:

We normally describe a continuous function as one whose graph can be drawn without any jumps. That's a good place to start, but is misleading.

An example of a well behaved continuous function would be $f \left(x\right) = {x}^{3} - x$

graph{x^3-x [-2.5, 2.5, -1.25, 1.25]}

In fact any polynomial is well defined everywhere and continuous.

A less obvious example of a continuous function is $f \left(x\right) = \tan \left(x\right)$

graph{tan(x) [-10, 10, -5, 5]}

This appears to be discontinuous, with 'jumps' at $x = \frac{\pi}{2} + n \pi$ but those values of $x$ are excluded from the domain.

Similarly, the following function is continuous on its domain $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$

$f \left(x\right) = \frac{x}{\left\mid x \right\mid}$

graph{x/abs(x) [-5, 5, -2.5, 2.5]}

If we add a definition of $f \left(0\right)$ then this becomes a discontinuous function.

$f \left(x\right) = \left\{\begin{matrix}\frac{x}{\left\mid x \right\mid} & \text{if x != 0" \\ 0 & "if x = 0}\end{matrix}\right.$

graph{(y-x/abs(x))(x^2+y^2-0.002) = 0 [-5, 5, -2.5, 2.5]}

This $f \left(x\right)$ fails the condition for continuity at the point $x = 0$.

${\lim}_{x \to 0} f \left(x\right)$ is not defined, let alone equal to $f \left(0\right)$