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..

Note that $$f(0)+f(2\cdot 0)=f(0)+f(0)=0\implies f(0)=0.$$
Consider $x_0\ne 0.$ It is
$$f(x)+f(2x)=0\underbrace{\implies}_{x=x_0/2} f(x_0/2)=-f(x_0),$$
$$f(x)+f(2x)=0\underbrace{\implies}_{x=x_0/4} f(x_0/4)=-f(x_0/2)=f(x_0).$$
By induction, $f(x_0/2^n)=-f(x_0)$ if $n$ is odd and $f(x_0/2^n)=f(x_0)$ if $n$ is even. Since $f$ is continuous at $0$ it is:
$$0=f(0)=\lim_{n\to\infty} f(x_0/2^{2n})=f(x_0).$$
Thus, $f\equiv 0$ is the only solution.
Thus 
