Points of Discontinuity

Below are some graphs illustrating examples of these various types of discontinuity, along with the reason why x=a is a point of discontinuity for the graphed function.  The depicted functions are continuous everywhere, excepting the one discontinuity shown. 

(i) The function isn’t defined at a.

(ii) The limit doesn’t exist at a.

(iii) The function doesn’t equal its limit at a.

The discontinuities in (i) and (iii) are called removable discontinuities, since one could redefine the function at x=a in such a way as to make the function continuous there.   The discontinuity in (ii) is called a jump discontinuity.

Some other types of discontinuity are illustrated in the following graphs: 

Oscillating discontinuity at 0.

Infinite discontinuity at 0.

Next, we’ll look at a couple of very strange examples (we won’t be using these examples in the rest of the course).

The first is a function that has a jump discontinuity at every point: 

 

The following function is continuous only at x=0, and has jump discontinuities at all other points: