algebra precalculus – Two different possibilities of a function satisfying $f(1)=f(0)+1$

Let $f: mathbb{R} rightarrow mathbb{R}$ be a differentiable function such that $f(1)=f(0)+1$. Then which of the following is/are true?

(a) $f^{prime}$ is constant.

(b) $f(2)=f(1)+1$

(c) $f^{prime}(x)=1$ for some $x$ in $(0,1)$ .

(d) $left|f^{prime}(x)right| leq 1$ for all $x$ in $(0,1)$

Source: Indian competitive exam for M.Sc in Applied Mathematics.

We see that $f(x)=x+C$, where $C$ is a constant satisfies the hypothesis of the problem.Hence all the options will be true.

On the other hand $f(x)=sinleft(frac{pi x}{2}right)$ also satisfies the hypothesis of the problem. But this function satisfies only option $(c)$.