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 |
Not Continuous |
Not Continuous |
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(hole) | (jump) | (vertical asymptote) |
Try these different functions so you get the idea:
Domain
A function has a Domain. In its simplest form the domain is all the values that go into a function. |
A function might be continuous or not, depending on its Domain!
Example: 1/(x-1)
At x=1 we have:
So there is a "discontinuity" at x=1
f(x) = 1/(x-1) over all Real Numbers |
g(x) = 1/(x-1) for x>1 | |
NOT continuous | Continuous |
g(x) does not include the value x=1, so it is continuous.
So 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:
"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) |
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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!
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:
which looks like: |
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:
which looks like: |
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.)