0
$\begingroup$

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 $q<1/\epsilon$

  • 0
    Is the edit OK? Note it is "Riemann" and "epsilon".2012-10-24
  • 2
    The Riemann function is a step function? If it were, you could just take $k$ to be $f$. Anyway, it's not good to just dump questions here without telling us what you know about the question, what you have tried, where you get stuck, and so on.2012-10-24

1 Answers 1