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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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    @DavidMitra Ahh sorry, I thought of a simple function.2012-03-22

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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.