@Jay
Wow, this is indeed interesting.
I mean theoretically it is clear that \( h(x):=f^{-1}(g(x))-x \) must have period 1 for two (strictly increasing continuous) solutions \( f \) and \( g \) of \( \phi(x+1)=F(\phi(x)) \).
Because \( h(x+1)=f^{-1}(g(x+1))-(x+1)=f^{-1}(F(g(x))-(x+1)=f^{-1}(g(x))+1-(x+1)=h(x) \).
But to see it as graphs is another thing .
And I would think that it is in no way a sinus wave. However the periodic analytic functions can built by Fourier series.
Wow, this is indeed interesting.
I mean theoretically it is clear that \( h(x):=f^{-1}(g(x))-x \) must have period 1 for two (strictly increasing continuous) solutions \( f \) and \( g \) of \( \phi(x+1)=F(\phi(x)) \).
Because \( h(x+1)=f^{-1}(g(x+1))-(x+1)=f^{-1}(F(g(x))-(x+1)=f^{-1}(g(x))+1-(x+1)=h(x) \).
But to see it as graphs is another thing .
And I would think that it is in no way a sinus wave. However the periodic analytic functions can built by Fourier series.