Calculating Dini derivative

Question

Let $f(x) = 1$ for rational numbers $x$ and $f(x)= -1$, otherwise.

Find all the Dini derivatives of $f$ at any rational $x$.

Answer 1

It's fairly straightforward, using the fact that $\mathbb Q$ is dense in $\mathbb R$ and the fact that ${\mathbb R} - {\mathbb Q}$ is dense in $\mathbb R,$ to show that at each $x \in {\mathbb Q}$ we have $D_{-}f(x) = 0$ and $D^{-}f(x) = +\infty$ and $D_{+}f(x) = -\infty$ and $D^{+}f(x) = 0.$

Imagine yourself located at a point on the graph where $x \in {\mathbb Q}.$ That is, imagine you're at a point whose coordinates are $(x,1),$ where $x \in {\mathbb Q}.$ Now imagine looking to your left at points on the graph. Arbitrarily close to you on the left there will be points at your height, but no points higher than you, so $D_{-}f(x) = 0.$ (Draw a sketch to guide your thinking.) Also, arbitrarily close to you on the left there will be points vertically lower than you by $2$ units, so their difference quotients relative to the point $(x,1)$ has no upper bound, and hence $D^{-}f(x) = +\infty.$ Argue in the same way for the right Dini derivates.