The Cantor function or devil's staircase is a function $f:[0,1] ightarrow[0,1]$ constructed as follows: On the mid-third $(1 / 3,2 / 3)$ it has the value $1 / 2$, on the mid-third of the first remaining interval $[0,1 / 3]$, i.e. on $(1 / 9,2 / 9)$, it has the value $1 / 4$, on $(7 / 9,8 / 9)$ it has the value $3 / 4$ and so on. This defines the function on all of $[0,1]$ except the Cantor set which remains after removing all the mid-thirds recursively. On the Cantor set we define $f$ in such a way that it will be a non-decreasing function. a) Show that $f$ is continuous. b) Show that $f^{\prime}=0$ on a set $S \subset[0,1]$ of measure 1 . Hence conclude that $f$ is not absolutely continuous.