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).

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)

So there is a "discontinuity" at 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:

limx→c f(x) = f(c)

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

"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:

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

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.

Example: How about the absolute value piecewise function:

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)