# 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 $$(c)$$.