Disproving a function is a step function.

Question

The function $f:[0,1] \to \mathbb{R}$ where $f(x) := 0$ if$ x \notin \mathbb{Q}$ and $f(p/q) : = \frac{1}{q} , q > 0, p,q$ coprime. How would you show that this is not a step function? Thanks!

Answer 1

This is known as Thomae's function. It is discontinuous at every rational number, and so is not continuous on any nontrivial interval -- which ought to contradict whatever definition of "step function" you're working with.