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could you help me with following problem?

I need to find two non-periodic functions "f" and "g" where their composition f(g) will be periodic.

Note that constant function is periodic too.

Thanks for help

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1 Answer 1

Consider the functions $f(x) = \sin(\mathrm{sgn}(x)x^2)$ and $g(x)=\mathrm{sgn}(x)\sqrt{|x|}$, where $\mathrm{sgn}(x)$ is $1$ if $x\geq 0$ and $-1$ otherwise. These can easily be shown to be non-periodic (hint for $f$: the zeroes get arbitrarily close together). But $f(g(x))=\sin(x)$, which is certainly periodic.

Edit: I didn't want to give a trivial example, but in light of your edit here it is. Let $f(x)=\mathrm{sgn}(x)$ and $g(x)=x^2$, which are non-periodic. Then $f(g(x))=1$.

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