Basic Analysis I — Real Analysis · Problem 3.2.16

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · continuous-functions

Problem

(Challenging) Suppose

f : \mathbb{R} \to \mathbb{R}

is continuous at

0

and such that

f(x + y) = f(x) + f(y)

for every

x

and

y

. Show that

f(x) = ax

for some

a \in \mathbb{R}

. Hint: Show that

f(nx) = nf(x)

, then show

f

is continuous on

\mathbb{R}

. Then show that

f(x)/x = f(1)

for all rational

x