The Cantor function is continuous on $[0,1]$

The Cantor function $c : [0,1] \to \mathbb{R}$ (The Cantor function on $[0,1]$, defined on the Cantor set through ternary digits and extended constantly across each removed interval) is continuous on $[0,1]$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point). It is moreover nondecreasing (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences), with $c(0) = 0$ and $c(1) = 1$.

No intermediate value theorem is used. The Cantor function is surjective onto $[0,1]$ by construction (The Cantor function is well defined, satisfies $c(x) \le c(y)$ whenever $x \le y$, is surjective onto $[0,1]$, and is constant on every interval removed from the Cantor set, claim 3), so its image is order-convex without any appeal to continuity, and continuity is then read off the monotone-with-interval-image criterion (A function on an interval satisfying $f(x) \le f(y)$ whenever $x \le y$, whose image is order-convex, is continuous). The implication runs in the direction opposite to the usual one: here surjectivity is known first and continuity is deduced.

• The Cantor function is well defined, satisfies $c(x) \le c(y)$ whenever $x \le y$, is surjective onto $[0,1]$, and is constant on every interval removed from the Cantor set deliberately claims nothing about continuity, and says so, for want of a definition of continuity at that point in the reading order. The present corollary supplies it, using nothing about $c$ beyond claims 2 and 3 of that theorem.

• The Cantor function is not strictly monotone. It is constant on every interval removed in the construction of the Cantor set (The Cantor function is well defined, satisfies $c(x) \le c(y)$ whenever $x \le y$, is surjective onto $[0,1]$, and is constant on every interval removed from the Cantor set, claim 4), so it is nondecreasing but not increasing, and in particular it is not injective. The continuous inverse theorem (Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f[I]$, and the inverse $g : f[I] \to I$ is continuous and strictly monotone in the same sense as $f$) therefore does not apply to it, and nothing here suggests otherwise.