How to find a differentiable map $T : \mathbb{R} \rightarrow \mathbb{R}$ whose fixed points are exactly integers?
I have to
a)Find points $|T'(x)| > 1$ (if any)
and prove that
b)There must exist at least one $x$ such that $|T'(x)| > 1$.
I think the functions $$x - Tx = x(x^2 - 1)(x^2 - 4)(x^2 - 9)...$$ and $$x - Tx = x(e^x - e^1)(e^{-x} - e^1)(e^x - e^2)(e^{-x} - e^2)(e^x - e^3)...$$ will satistfy the criteria.
Is there any closed form expressions for $T'(x)$?
How can I proceed further?