Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Define $g(0) = f(0)$ and $g(x) = \max\{f(y) \mid 0 \leq y \leq x\}$ for $0 < x \leq 1$. Show that $g$ is well-defined and that $g$ is a monotone continuous function.
Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Define $g(0) = f(0)$ and $g(x) = \max\{f(y) \mid 0 \leq y \leq x\}$ for $0 < x \leq 1$. Show that $g$ is well-defined and that $g$ is a monotone continuous function.