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?