# Continuous Functions

A function is continuous when its graph is a single unbroken curve ...

... that you could draw without lifting your pen from the paper.

That is not a formal definition, but it helps you understand the idea.

Here is a continuous function:

## Examples

So what is **not continuous** (also called **discontinuous**) ?

Look out for holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).

**Not**Continuous

(hole)

**Not**Continuous

(jump)

**Not**Continuous

(vertical asymptote)

Try these different functions so you get the idea:

(Use slider to zoom, drag graph to reposition, click graph to re-center.)

## Domain

A function has a Domain.

In its simplest form the domain is all the values that **go into** a function.

We may be able to choose a domain that makes the function continuous

### Example: 1/(x−1)

At x=1 we have:

So there is a "discontinuity" at x=1

f(x) = 1/(x−1)

So f(x) = 1/(x−1) over **all Real Numbers** is NOT continuous

Let's change the domain to **x>1**

g(x) = 1/(x−1) for **x>1**

So g(x) IS continuous

In other words g(x) does **not** include the value x=1, so it is **continuous**.

When a function is **continuous within its Domain**, it is a continuous function.

## More Formally !

We can define **continuous** using Limits (it helps to read that page first):

A function **f** is continuous when, for **every** value **c** in its Domain:

f(c) is defined,

and

*lim***x→c***f(x) = f(c)*

*"the limit of f(x) as x approaches c equals f(c)*"

The limit says:

"as x gets closer and closer to c

then f(x) gets closer and closer to f(c)"

And we have to check from both directions:

as x approaches c (from left) then f(x) approaches f(c) |
||

AND as x approaches c (from right) then f(x) approaches f(c) |

If we get different values from left and right (a "jump"), then the limit does not exist!

And remember this has to be true for every value **c** in the domain.

## How to Use:

Make sure that, for all **x** values:

**f(x)**is defined- and the limit at
**x**equals**f(x)**

Here are some examples:

### Example: f(x) = (x^{2}−1)/(x−1) for all Real Numbers

The function is **undefined** when x=1:

(x^{2}−1)/(x−1) = (1^{2}−1)/(1−1) = **0/0**

So it is **not** a continuous function

Let us change the domain:

### Example: g(x) = (x^{2}−1)/(x−1) over the interval x<1

**Almost** the same function, but now it is over an interval that does **not** include x=1.

So now it **is** a continuous function (does not include the "hole")

### Example: How about this piecewise function:

It looks like this:

It is **defined** at x=1, because **h(1)=2** (no "hole")

But at x=1 **you can't say what the limit is**, because there are two competing answers:

- "2" from the left, and
- "1" from the right

so in fact the limit does not exist at x=1 (there is a "jump")

And so the function is **not continuous**.

But:

### Example: How about the piecewise function absolute value:

At x=0 it has a very pointy change!

But it is still **defined** at x=0, because **f(0)=0** (so no "hole"),

And the limit as you approach x=0 (from either side) is also **0** (so no "jump"),

So it is in fact ** continuous**.

(But it is not differentiable at x=0)