Proving certain almost homogeneous function is linear

Question

I have the following problem:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function satisfying $ f(3x) = 3f(x) $ and differentiable at $ x = 0 $. We are asked to show $ f $ is linear.

If it were a general homogeneous function, I could show this, but now I can't. All I know is that $ f'(0) = f(1) $ and $ f(0) = 0 $. Can someone please show me the rest? Thanks.

Answer 1

By induction one can show that $f(x)=3^nf(\frac{x}{3^n})$. Now we put $h=\frac{x}{3^n}$ then $f(x)=x\frac{f(h)}{h}$. Since $f$ is differentiable a 0, $\frac{f(h)}{h}\longrightarrow f’(0):=a$ when $h\longrightarrow 0$. The claim follows.

Answer 2

Assuming differentiability at $0$, we don't actually need continuity of $f$. Since $f(0)=0$, we have for all $x\ne 0$, $$ f(x)=3f\left(\frac x 3\right)=\cdots=3^nf\left(\frac x {3^n}\right)=x\frac{f\left(\frac x {3^n}\right)-f(0)}{\frac x{3^n}}\xrightarrow{n\to\infty}xf'(0), $$ that is, $f(x)=cx$ for some $c\in\Bbb R$.