MathNet 2018-part-05 — 181 problems (EN–VI) Question 24

Two continuous functions $f, g: [0, 1] \to [0, 1]$ are such that $f \circ g = g \circ f$, and one of them is nondecreasing. Prove that there exists $a \in [0, 1]$ so that $f(a) = a = g(a)$.