# Riemann Integrability of Composition

1. Apr 11, 2013

### Shoelace Thm.

1. The problem statement, all variables and given/known data
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?

2. Relevant 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.

3. The attempt at a solution

2. Apr 11, 2013

### Shoelace Thm.

Any suggestions?

3. Apr 11, 2013

### 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?

4. Apr 11, 2013

### 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.