What are they?
When you’re looking at the behavior of a function at a jump discontinuity, you already know that the general limit doesn’t exist
In this video we’re talking about jump discontinuities, or 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.
Skip to section:
0:18 // What are jump discontinuities? What are they compared to infinite/essential discontinuities?
0:52 // The general limit will never exist at a jump discontinuity
1:59 // What’s going on at x=-2?
4:46 // What’s going on at x=3?
7:05 // What’s going on at x=5?
8:22 // How to find the value of the function at a jump discontinuity
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.