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..

$f(x)=0$ is the only solution.
$$f(0)=-f(2\cdot 0) \iff f(0)=0$$
Suppose that for any $x$,  $f(x) = \epsilon$. But this means that for any $\delta$ and for large $n$ we have:
$$\frac{x}{2^n} < \delta$$
$$\large f\bigg(\frac{x}{2^n}\bigg)= (-1)^n\cdot\epsilon$$
But supposing $f$ was continuous, we could choose a $\delta$ small enough that $|f(x)|<\epsilon$ for $x< \delta$. Thus if this function is continuous, it can have no values other than $0$.
