We take a look at the various types of discontinuity and how they occur.
A removable discontinuity is a point where the function has a
discontinuity, but may be redefined at that point to make it continuous.
This is where the function has a jump either side of the point .
You can see that this function has a discontinuity at (in fact this
point is not in the domain of ) however if we define , then
this becomes a continuous function.
The function has the given graph. It has a jump discontinuity at
. Notice that there is no way to redefine at to make it
continuous as in the previous example.
A function has an infinite discontinuity if the limit at is plus or
An oscillating discontinuity occurs when the value of the function is
changing so rapidly that a limit is not possible. The classic example is
On what intervals are the following functions continuous? Beware of
removable discontinuities which this program will ignore!
Are the following functions continuous at the given point?