Coloring F(x) = F(exp(x))

[tommy1729]

I corrected the title and first post.

Im a bit confused by this :
If F is piecewise analytic and
F(x) = F(exp(x))
Then
F(x) - F(exp(x)) = g(x)
And g(x) should be piecewise analytic too.

But g(x) = 0 locally.
However 0 has no analytic continuation apart from being 0 EVERYWHERE.
This seems like a contradiction.

I might have asked this before srr.

Regards

Tommy1729