A composition of a periodic function that is somehow not periodic.

Question

Is there some g for which g(x mod 1) is not periodic? If so, please tell me what it is.

Answer 1

You may already know this, but I want to write it out in detail - here is a counterexample to your conjecture:

Let $f(x)=(x$ mod $1)-{1\over 2}$. Note that $f$ is periodic of period $1$, and so is its integral: $\int_a^bf(x)dx=\int_a^{b+1}f(x)dx$, since $\int_b^{b+1}f(x)dx=0$ for all $b$.

Similarly, by what I wrote below, $g\circ f$ is periodic of period $1$ for all functions $g$.

But $xf(x)$ is not periodic - one way to see this is to note that $xf(x)$ is bounded on every bounded interval, but is unbounded in general (for $z$ a positive integer, we have $({3\over 4}+z)f({3\over 4}+z)>{z\over 4}$ which goes to infinity as $z\rightarrow\infty$).

So there is no $g$ such that $xf=(\int f)+g\circ f$, since otherwise $xf$ would be periodic as it is the sum of two periodic functions with the same period.

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There is no such $g$.

In general, if $f$ is periodic with period $a$ and $g$ is any function, then $g\circ f$ is periodic with period $a$, since $$(g\circ f)(x+a)=g(f(x+a))=g(f(x))=(g\circ f)(x).$$

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EDIT: the OP asks in a comment about infinite compositions, and infinite sums. The result still holds, and the key fact is the following:

If $h=\lim_{i\rightarrow\infty}g_i$ (that is, $h$ is the pointwise limit of the $g_i$s), and each $g_i$ is periodic with period $a$, then $h$ is periodic with period $a$.

Proof: $$h(x+a)=\lim_{i\rightarrow\infty}g_i(x+a)=\lim_{i\rightarrow\infty}g_i(x)=h(x).$$

How is this relevant? Well, let $g$ be arbitrary, and $f$ have period $a$. Then $g^\infty\circ f=\lim_{i\rightarrow \infty}g\circ g\circ g\circ . . .\circ g\circ f$. Well, by the fact above the fold + induction, each $g^n\circ f$ is periodic with period $a$; so $g^\infty\circ f$ is the pointwise limit of periodic-with-period-$a$ functions, hence has period $a$. Similarly for infinite series.

(Now maybe you don't define infinite compositions via pointwise limits; in that case, how do you define them?)