Show that f is not an increasing function on any neighbourhood of 0.

Question

Let $f:\mathbb{R} \to \mathbb{R}$ be defined as $$ f(x)= \begin{cases} x, & x \in \mathbb{Q}\\ x−x^2, & x \not \in \mathbb{Q} \end{cases} $$ Show that $f$ is not an increasing function on any neighbourhood of $0$.

Please help I have no idea how to even start this proof!

Answer 1

HINT

Note that if $x \in (0,\epsilon)$, then $x^2>0$ and $x^2<x$. Therefore, $x-x^2 < x$, so you only have to find a rational $x$ and a slightly larger irrational $x+\delta$.