# 7. Continuous and Discontinuous Functions

by M. Bourne

This section is related to the earlier section on Domain and Range of a Function.
There are some functions that are not defined for certain values of *x*.

## Continuous Functions

Consider the graph of *f*(*x*) = *x*^{3} − 6*x*^{2} − *x* + 30:

Graph of `y=x^3 - 6x^2 - x + 30`, a continuous graph.

We can see that there are no "gaps" in the curve. Any value of *x* will give us a corresponding value of *y*. We could continue the graph in the negative and positive directions, and we would never need to take the pencil off the paper.

Such functions are called **continuous functions**.

## Functions With Discontinuities

Now consider the function `f(x) = 1/(x-1).`

We note that the curve is not **continuous** at `x =
1`.

Graph of `y=1/(x-1)`, a discontinuous graph.

We observe that a **small change** in *x*
near `x = 1` gives a very **large change** in the value of
the function.

For a function to be **continuous** at a point, the
function must exist at the point and any small change in *x*
produces only a small change in `f(x)`.

In simple English: The graph of a **continuous
function** can be drawn **without lifting the pencil
from the paper.**

Many functions have **discontinuities** (i.e. places where
they cannot be evaluated.)

### Example

Consider the function

`f(x)=2/(x^2-x)`

Factoring the denominator gives:

`f(x)=2/(x^2-x)=2/(x(x-1))`

We observe that the function **is not defined** for `x = 0` and `x = 1`.

Here is the graph of the function.

Graph of `f(x)=2/(x^2-x)`, a discontinuous function.

We see that small changes in *x* near 0 (and near 1) produce
large changes in the value of the function.

We say the function is **discontinuous **when *x* = 0 and *x* = 1.

There are 3 **asymptotes** (lines the curve gets closer to, but doesn't touch) for this function. They are the `x`-axis, the `y`-axis and the vertical line `x=1` (denoted by a dashed line in the graph above).

### Using computers to draw discontinuous graphs

**Note:** You will often get strange results when using
Scientific Notebook (or any other mathematics
software) if you try to graph functions which have
discontinuities.

Here is the same function `f(x)=2/(x^2-x)` in the default graph view in Scientific Notebook:

It is showing us all the vertical values that it can (from an extremely small negative number to a very large positive number) - but we can't see any detail (certainly none of the curves).

We need to restrict the *y*-values so we can see the true
shape of the curve, like this (I have changed the view of the
vertical axis from -12 to 10):

### Continuity and Differentiation

Later you will meet the concept of differentiation. We will learn that a function is differentiable only where it is continuous.