Calculus How To

Jump Discontinuity (Step)/Discontinuities of the First Kind

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A jump discontinuity (also called a step discontinuity or discontinuity of the first kind) is a gap in a graph.

The following graph jumps at the origin (x = 0).

In order for a discontinuity to be classified as a jump, the limits must:

  1. exist as (finite) real numbers on both sides of the gap, and
  2. cannot be equal. If the limits are equal, it’s a hole, not a jump (more formally, holes are called removable discontinuities).

The difference between the two limits is the jump at that point (Sohrab, 2003). Surprisingly, the number of jumps in any particular function are countable; In other words, it’s not possible to have an infinite number of jumps, even in continuous functions (Sohrab, 2003).


When do Jumps Happen?

A jump discontinuity usually only happens in piecewise or step functions.

Piecewise functions are defined on a sequence of intervals.

step function example

Example of a step function (the red graph). This particular step function is right-continuous.


Step functions are a sub-type of piecewise functions, where there’s a series of identical “staircase” steps.
piecewise function

Notation for Jump Discontinuities

In notation, a jump discontinuity can be defined in terms of limits on either side of the jump. Let’s say you have a function, f(t), which has a jump discontinuity at t = 10. The following notation describes the jump:

Left limit:


Right limit:

The jump itself can be defined in terms of the two limits:
f (10 +) – f (10 -).

Jump vs. Step

Although “step discontinuity” is a fairly common term, it tends to be an informal one. The usual name for this type of discontinuity is a jump discontinuity. However, when it looks like a physical step, it makes sense to call it that (rather than a jump, which would bring to mind a large gap in the horizontal axis, which isn’t always the case!). Which term you use is usually a matter of personal choice, or the choice of your instructor.

Examples of a Step Discontinuity

The function


has a jump discontinuity when x is one. There is no single limit at this point; even though the one sided limits L and L+ both exist, because they are not equal. If you imagined walking along the curve, you would have to do some serious jumping when you got to one. This is illustrated below.
step discontinuity

In the graph below, there is a step discontinuity at -4, because the left and right hand limits both exist and are not infinite but are different. At 2 there is another step discontinuity; the right limit is -1 and the left limit is 5. The discontinuity at 4, however, is not a step discontinuity because the left and right hand limits are equal. This is another type of discontinuity called a removable discontinuity.


References

Bogley, William A. Jump Discontinuities. Stage 4, Calculus Quest. Oregon State Math 251. Retrieved from https://oregonstate.edu/instruct/mth251/cq/Stage4/Lesson/jumps.html on March 24, 2019
Sohrab, H. (2003). Basic Real Analysis. Springer Science and Business Media.
Thomson, B. et al., (2008). Elementary Real Analysis, Volume 1. ClassicalRealAnalysis.com.
Wineman, A. & Rajagopal, K. (2000). Mechanical Response of Polymers: An Introduction. Cambridge University Press.


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