Does $f'(x)\in \mathbb Z$ a.e. implies that $f$ is an affine function?

real-analysis analysis derivatives measure-theory

Yes, $f'$ must be constant. This has been answered in the comments; the result in https://mathoverflow.net/questions/266377/how-quickly-can-the-derivative-of-an-everywhere-differentiable-function-change-s gives a kind of positive measure Darboux theorem. This is a special case of How irregular can $f'$ be beyond Darboux's Theorem?

Related

Control over bump function's second derivative
Is there any continuous surjective map from $\mathbb{R}^3$ to $\mathbb{R}^3$ s.t. for every $x$ in $\mathbb{R}^3$ $f(x) \cdot x= 0$?
Number of real zeroes of iterated polynomial: $x^3-2x+1$
2D Random Walk Hitting Time
Can octonions be represented by infinite matrices?
Proving that $\int_0^1 \frac{\ln(x)+\ln(\sqrt x+\sqrt {1+x})}{\sqrt {1-x^2}} dx=0$
Topology on the space of paths
Groups with polynomial growth rate
Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?
What lessons have mathematicians drawn from the existence of non-standard models?
Any proof to $\pi^{e}$'s irrationality?
zeros of exponential polynomials