Question 19
a) Show that there exist functions $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ such that $f \circ g = g \circ f$, $f \circ f = g \circ g$ and $f(x) \ne g(x)$, for all $x \in \mathbb{R}$.
b) Prove that if $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ are continuous and have the properties $f \circ g = g \circ f$ and $f(x) \ne g(x)$, for any $x \in \mathbb{R}$, then $(f \circ f)(x) \ne (g \circ g)(x)$, for all $x \in \mathbb{R}$.
b) Prove that if $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ are continuous and have the properties $f \circ g = g \circ f$ and $f(x) \ne g(x)$, for any $x \in \mathbb{R}$, then $(f \circ f)(x) \ne (g \circ g)(x)$, for all $x \in \mathbb{R}$.