Paper 1, Section II, D

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a function that is continuous at at least one point $z \in \mathbb{R}$. Suppose further that $g$ satisfies

$g(x+y)=g(x)+g(y)$

for all $x, y \in \mathbb{R}$. Show that $g$ is continuous on $\mathbb{R}$.

Show that there exists a constant $c$ such that $g(x)=c x$ for all $x \in \mathbb{R}$.

Suppose that $h: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function defined on $\mathbb{R}$ and that $h$ satisfies the equation

$h(x+y)=h(x) h(y)$

for all $x, y \in \mathbb{R}$. Show that $h$ is either identically zero or everywhere positive. What is the general form for $h$ ?

*Typos? Please submit corrections to this page on GitHub.*