What is a jump discontinuity?
Jump discontinuities are common in piecewise-defined functions
You’ll usually encounter jump discontinuities with piecewise-defined functions, which is a function for which different parts of the domain are defined by different functions. A common example used to illustrate piecewise-defined functions is the cost of postage at the post office.
Below is an example of how the cost of postage might be defined as a function, as well as the graph of the cost function. They tell us that the cost per ounce of any package lighter than ???1??? pound is ???20??? cents per ounce; that the cost of every ounce from ???1??? pound to anything less than ???2??? pounds is ???40??? cents per ounce; etc.
???f(x)=\begin{cases}0.2 & \quad 0<x<1\\ 0.4 & \quad 1\leq x<2\\ 0.6 & \quad 2\leq x<3\\ 0.8 & \quad 3\leq x<4\\ 1.00 & \quad 4\leq x \end{cases}???
Every break in this graph is a point of jump discontinuity. You can remember this by imagining yourself walking along on top of the first segment of the graph. In order to continue, you’d have to jump up to the second segment.
In this video we'll look at what jump discontinuities are and how to find them
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When you’re looking at the behavior of a function at a jump discontinuity, you already know that the general limit doesn’t exist, that the function isn’t continuous, and also that it’s not differentiable. Which means that all you really have left to investigate are the one-sided limits, and the actual value of the function at that point.
The one-sided limits will never be equal if it’s a jump discontinuity, so you want to look at the left-hand limit by tracing the function from the left, or negative side, and identifying the y-value where you run out of graph. You also want to trace the function from the right, or positive side, and identify the y-value where you run out of graph. This will give you the left- and right-hand limits, respectively.
You'll also want to see where the graph has a "filled in" circle. It might be connected to the left piece of the graph, or to the right piece of the graph, or it might be floating somewhere else along the vertical line where the jump discontinuity exists. Regardless of where it is, the filled in circle represents the function's actual value at that point.
Jump discontinuities are also called "discontinuities of the first kind." These kinds of discontinuities are big breaks in the graph, but not breaks at vertical asymptotes (those are specifically called infinite/essential discontinuities). You’ll often see jump discontinuities in piecewise-defined functions. A function is never continuous at a jump discontinuity, and it’s never differentiable there, either.