# Discontinuities

We take a look at the various types of discontinuity and how they occur.

## Removable Discontinuity

A removable discontinuity is a point where the function has a discontinuity, but may be redefined at that point to make it continuous.

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.

## Jump Discontinuity

This is where the function has a jump either side of the point .

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.

## Infinite Discontinuity

A function has an infinite discontinuity if the limit at is plus or minus infinity.

## Oscillatiing Discontinuity

An oscillating discontinuity occurs when the value of the function is changing so rapidly that a limit is not possible. The classic example is

## Exercises

On what intervals are the following functions continuous? Beware of removable discontinuities which this program will ignore!

1. Are the following functions continuous at the given point?
2. at
3. at
4. at

Robert Judd
2001-11-24