Counter example of a given two sets .

Question

let $f:(a, b) \rightarrow \Bbb R$ be a differentiable map, and define two sets, $A=\left\{f^{\prime}(x): x \in(a, b)\right\}$ and $B=\left\{\frac{f(y)-f(x)}{y-x}: x \neq y \in(a, b)\right\}$ I know that $\sup B$ and $\inf A$ are different. I would like to find $\sup B$ and $\inf A$. I am also trying to think about and counterexample to analyse here. Another option would try by contradiction that $\sup B$ and $\inf A$ are different.

Answer 1

You don't know that $\sup B \neq \inf A$. Define $f: (a, b) \to \Bbb R$ to be a constant function. Then $A=B= \{ 0 \}$.