Jump Discontinuities

If f(x) fails to be continuous because

 lim f(x) x --> a

fails to exist (or is infinite), then there is no way to remove the discontinuity - the limit statement takes into consideration all of the infinitely many values of f(x) sufficiently close to a and changing a value or two will not help. We call this an essential discontinuity.

If f(x) is discontinuous at x=a because
 lim f(x) x --> a
fails to exist or is infinite, then f(x) has an essential discontinuity at x=a. If a discontinuity is not removable, it is essential.

Types of Essential Disontinuities

Essential discontinuities, and their corresponding pictures, come in different flavors according to the manner in which the

 lim f(x) x --> a

fails to exist or be finite. The possibilities are:

1. The left- and right-hand limits exist and are finite but do not agree.

2. f(x) has a vertical asymptote at x=a, so the one-sided limits are infinite.

3. The limit REALLY does not exist, as in the case of

 lim sin(/x) x --> 0

1. Jump Discontinuities

If the left- and right-hand limits at x=a exist but disagree, then the graph jumps at x=a.

Thus the function

 { 1 if u > 1, s(u) = u/|u| = undefined if u = 0, -1 if u < 1.

(introduced by Andron's Uncle Smith) has a jump discontinuity at u=0.

As other examples, the functions h(t) and j(t) from "Left- and Right-hand Limits" in Stage 3 have jump discontinuities.

/Stage4/Lesson/jumps.html