The importance of continuity of functions in proving some properties of convolution. What is the importance of continuity of functions in proving the commutative and associative properties of convolution if the functions are assumed to be Riemann integrable $2\pi$-periodic functions ?
Could anyone clarify this for me? As the book "Fourier Analysis" by Stein and Shakarchi assumes that the functions are continuous at the beginning of the proof in page 45.
 A: In the proof of (v), they use that $g$ is continuous in order to deduce that it is uniformly continuous on $[-\pi, \pi]$ and bound $|g(x_1 - y) - g(x_2 - y)|$ by $\epsilon$. The continuity of $f$ is not used here, only its boundedness (by $B$), which would also result from the weaker assumption of Riemann integrability.
In the proof of (iii) they use the change of variable theorem, which is stated (without proof) on page 292 (theorem 2.2 in the appendix) only for continuous functions. The truth is that the theorem is valid for a larger class of functions (Riemann-integrable ones, for instance), so it is a mystery why the authors chose a weaker statement.
The proof of (vi) depends on Fubini's theorem. They don't state it in the book, there is only a version for continuous and moderately decreasing functions on $\Bbb R^d$ on page 295 (theorem 3.1 in the appendix). Again, Fubini's theorem holds for a vastly larger class of functions - why they chose to only use continuous ones is a mystery.
Finally, the proof of (iv) requires a change of variables and Fubini's theorem, which by the authors' choice are given only for continuous functions.
To conclude, the proofs of (iii), (iv) and (vi) don't need continuity if you are provided with versions of Fubini's theorem and the change of variable theorem for (Riemann or Lebesgue) integrable functions. The proof of (iv) needs only the continuity of $g$. Notice, though, that the continuity assumption is removed using lemma 3.2 on page 47, which allows one to reduce the case of merely integrable functions to the case of continuous ones. In any case, this is a strange choice of presentation of this material.
