Cantor function

Construction. On the middle-third interval $(1/3,2/3)$ removed at step $1$, set $F=1/2$. On $(1/9,2/9)$ set $F=1/4$, on $(7/9,8/9)$ set $F=3/4$, etc.

Properties.
- Continuous and non-decreasing.
- Constant on each removed interval.
- Surjective onto $[0,1]$.
- Derivative $0$ almost everywhere.

Cardinality. Surjection from the Cantor set onto $[0,1]$ confirms $|K|\ge2^{\aleph_0}$, hence $|K|=2^{\aleph_0}$.

Conclusion. The Cantor function provides a concrete surjection from the Cantor set to $[0,1]$.