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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!

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How do you define "step function"? – martini Mar 22 '12 at 13:05
Is $f$ constant on any interval? Does $f$ take on only finitely many values? – David Mitra Mar 22 '12 at 13:07
@DavidMitra Ahh sorry, I thought of a simple function. – AD. Mar 22 '12 at 13:12

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up vote 1 down vote accepted

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.

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