Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics II / 4 / a

A function $f:(a,b)\to \mathbb{R}$ is differentiable on $(a,b)$ if, for every $x\in (a,b)$, the finite limit of a functionexists. Here $h\ne 0$ and is sufficiently small that $x+h\in (a,b)$. This limit of a function is the derivative of $f$ at $x$.

The mean value theorem states: if a real-valued function $f$ is continuous on a closed interval $[u,v]$, where $u<v$, and differentiable on $(u,v)$, then there is $c\in (u,v)$ with

Take any $a<u<v<b$. By differentiability implies continuity, $f$ is continuous on $[u,v]$, and it is differentiable on $(u,v)$. The mean value theorem gives $c\in (u,v)$ for whichTherefore $\boxed{f(u)\le f(v)\text{ whenever }u<v}$: $f$ is a nondecreasing function, which is the non-strict meaning of increasing here.

For any $u<v$ in the interval, the mean value theorem, applicable by differentiability implies continuity, givesAll pairs of values are equal. Thus $\boxed{f\text{ is a constant function}}$, the constancy theorem.