Suppose f : R → R is continuous. Let λ be a positive real number, and assume that for every $x ∈ R$ and $a > 0, f(ax) = aλf(x).$
If λ = 1, show that f is differentiable at 0 if and only if it is linear.
I asked a similar question before. But this one is different.
I get $f'(0)=f(1)$ by manipulating the algebra. However, just because $f'(0)=f(1)$ doesn't mean it is linear. So, I need some hints on how to continue the proof.
Besides, since the question asks me to prove "if and only if", which means I'll also have to use "linear" as condition to deduce "$f$ is differentiable at $0$". Any hint on that?
It is another similar question I asked before Differentiability of an homogeneous and continuous function $f$ ($f(\alpha x)=\alpha^\beta f(x)$)