Finding Continuous Functions Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) + f(2x) = 0$ 
I'm thinking; 
Let $f(x)=-f(2x)$ 
Use a substitution $x=y/2$ for $y \in \mathbb{R}$. 
That way $f(y)=-f(y/2)=-f(y/4)=-f(y/8)=....$ 
Im just not sure if this is a good approach. Opinions please..

By induction we prove that
$$f(x)=(-1)^nf\left(\frac x{2^n}\right)$$
but by the sequential characterization of the continuity we have
$$f\left(\frac x{2^n}\right)\xrightarrow{n\to\infty}f(0)$$
so since for $x=0$ we get  $f(0)=0$ so $f(x)$ is constant 
$$f(x)=f(0)=0\quad\forall x\in\Bbb R$$
