How can I prove that the next function $f:[0,1] \to \mathbb{R}$ defined as follows is not monotone for any subinterval $[a,b] \subseteq [0,1]$? Some suggestions?
Let $(0,1)\cap\mathbb{Q} = \{r_1,r_2,\dotsc\}$ the rational numbers between 0 and 1 ordered in a sequence.\begin{equation}f(t):= \begin{cases}2^{-k} & \text{for} \quad t = r_k, \\0 & \text{otherwise}. \end{cases}\end{equation}