J is for . . .

Ready for today's Mad-Cool-Math nugget?

J is for jump discontinuity.

Here is a picture of a nice, smooth continuous function:

Image courtesy: Wikicommons

Compare that unbroken, blue swoop to this choppy, blue step function, which hops up at every number leaving a gap. Such a break is called a jump discontinuity.

Image courtesy: Wikimedia Commons

This particular step function is called a floor function. We've got two axes here: the horizontal x-axis and the vertical y-axis. Where x = 0, you see a solid blue dot. That means y = 0 there too. Where x = 0.5, y still equals 0. Where x = 0.9, y is still 0, and so on until you get to x = 1. Then the y value jumps up to 1 too.

Similarly, when x = 1.2 or 1.4 or 1.8 or 1.99999, the corresponding y is 1. Then when x = 2, y jumps from 1 up to 2.

What kind of kooky person uses something like this? You do, every time someone asks your age.