  Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}:
n \leq x\}$ on which the Euler totient function $\varphi(n)$ is non-decreasing.
We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) \pi(x)$ for all $x
\geq 10$, answering questions of Erd\H{o}s and Pollack--Pomerance--Trevi\~no. A
similar result is also obtained for the sum of divisors function $\sigma(n)$.
