I want to find a sequence of simple functions $\phi_\epsilon$ and $\psi_\epsilon (x)$ for the function f,
$$f(x)=\left\{ \begin{array}{ll} x & x\in Q^c\cap[0,1], \\ 0& x\in Q\cap[0,1]. \end{array} \right.$$
such that $\phi_\epsilon (x)\leq f(x) \leq \psi_\epsilon (x), \forall x\in [0,1] $ and $|\psi_\epsilon (x)-\phi_\epsilon (x)| < \epsilon$.
I tried to make a partition of the Range of f and find the sequence as left end point of subintervals of the partition to find $\phi_\epsilon$. But it is becoming complicated as the range is set of all irrationals in [0,1]and 0. This is the method from the "simple approximation lemma,Chapter 3, Real analysis(4th edition) by H L Royden (Page number 61)".
Can someone guide me?