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) = x3 − 6x2 − x + 30:
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 
.
We note that the curve is not continuous at x = 1.
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 
.
Factorising the denominator gives:
.
We observe that the function is not defined for x = 0 and x = 1.
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 continuous for all values of x except x = 0 and x = 1.
Note: You will often get strange results when using
Scientific Notebook (or LiveMath or any other mathematics
software) if you try to graph functions which have
discontinuities. Here is the same function 
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 need to restrict those 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.
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