Finding all differentiable functions satisfying a property

Question

I am trying to find all differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ s.t. $$ \forall x \in \mathbb{R}, \; \forall n \in \mathbb{N}^{+}, f'(x)=\frac{f(x+n)-f(x)}{n}$$ I know that a sufficient class of functions with this property is all linear functions $f(x)=ax+b$ for some real constants $a$ and $b$ since $$f'(x)=a=\frac{a(x+n)+b-(ax+b)}{n}=\frac{f(x+n)-f(x)}{n}$$ but what is the necessary class of functions with the property?

Answer 1

Since $f$ is differentiable and $f'(x) = f(x+1) - f(x)$, it follows that $f'$ is differentiable as well, with \begin{align} f''(x) = f'(x+1)-f'(x) &= (f(x+2)-f(x+1)) - (f(x+1)-f(x)) \\ &= f(x+2)-2f(x+1)+f(x) \\ &= (f(x+2)-f(x)) - 2(f(x+1)-f(x)) \\ &= 2f'(x) - 2f'(x) \\ &= 0. \end{align} Hence, $f'$ must be constant, i.e. $f$ must be linear.