I was thinking about the problem:
Let $f:\mathbb R \rightarrow \mathbb R$ be differentiable function such that $f'(x)$ is continuous and $f(x+1)=f(x)+1$ for all $x \in\mathbb R$. Then which of the following options is correct?
(a) $f'(x)$ must be bounded,
(b) $f(x)$ must be bounded,
(c) Both $f(x)$ and $f'(x)$ must be bounded,
(d) Both $f(x)$ and $f'(x)$ must be unbounded.
My attempts: If I take $f(x)$ to be of the simplest form that is $f(x)=x,$ so that the given condition is satisfied then we see that $f'(x)=1,$ which is bounded. So I think that choice (a) is the correct option. Am I going in the right direction? I want a proof in a more generalized way. Please help. Thanks in advance for your time.