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:
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.)
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
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,
limx→cf(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)
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) = (x2−1)/(x−1) for all Real Numbers
The function is undefined when x=1:
(x2−1)/(x−1) = (12−1)/(1−1) = 0/0
So it is not a continuous function
Let us change the domain:
Example: g(x) = (x2−1)/(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.
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)