This is Thomae's function. It is continuous at every irrational number, and discontinuous at every rational number. Yup, you read that right. There are an infinite number of discontinuities between zero and one, but it is continuous at even more places.
Thomae's function is a really fun way to confuse real analysis students... I should know, I'm one of them. Nothing about this function makes natural, intuitive sense. But, once you get into the math of it all, it starts to...
00:00 Intro
01:50 How does Thomae's Function Work?
03:48 What does it Mean to be Continuous?
07:25 So HOW is Thomae's Function Continuous?
10:51 Outro (and Apologies)