The Riemann function $f(x)=\begin{cases}\frac 1 q \text{ if } x=\frac p q \in\Bbb Q\\ 0\text{ if } x\in \mathbb R\setminus \Bbb Q\end{cases}$
is a step function. Then, for any epsilon construct a step function $k:[0,1]\to \mathbb R$ such that $||f-k||=\sup\{|f(t)-k(t)|:t \in [0,1]\}<\epsilon$
Suggestion: restrict to $q<1/\epsilon$