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