I am interested in knowing more about the continuity of the Cantor Function. I have read that the function is discontinuous precisely at the points of the Cantor Set and it is continuous everywhere else.
Can anyone please elaborate on why exactly this is true?
Hint:
The Cantor function $\phi:[0,1] \to [0,1]$ is monotone non-decreasing. Hence, there are at worst countably many jump discontinuities. However, $\phi$ is surjective and $\phi([0,1]) = [0,1]$.