Riemann Integrability of Composition

[Shoelace Thm.]

Homework Statement

Let ψ(x) = x sin 1/x for 0 < x ≤ 1 and ψ(0) = 0.
(a) If f : [-1,1] → ℝ is Riemann integrable, prove that f \circ ψ is Riemann integrable.
(b) What happens for ψ*(x) = √x sin 1/x?

Homework Equations

I've proven that if ψ : [c,d] → [a,b] is continuous and for every set of measure zero Z \subset [a,b], ψ^{\text{pre}}(Z) is a set of measure zero in [c,d], then if f is Riemann integrable, f \circ ψ is Riemann integrable. However, this doesn't apply well in this situation. What can I do? I have a hunch that both f \circ ψ and f \circ ψ* are Riemann integrable.

The Attempt at a Solution

 

[Shoelace Thm.]

Any suggestions?

 

[micromass]

What do you know about Riemann integration?? What theorems could come in handy?
For example, do you know that a function is Riemann integrable iff the set of discontinuities has measure 0?

 

[Shoelace Thm.]

SammyS: My apologies, it was not twenty-four hours.

Micromass: I am aware of that theorem, and in fact used it to prove the theorem in the original post. The problem is that I do not know how to show that the set of discontinuities of f \circ \psi are of measure 0. Clearly it is discontinuous at a point u iff f is discontinuous at \psi(u), but I don't know where to go from here.