True or false:
If $g(x)=f(x)+f(2x)$ with $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, then $f$ is continuous.
My idea was to find a counterexemple since, first, I claim that this is false. I did not find an example. After that I suppose that $g(x)=0,\forall x\in \mathbb{R}$ and, by induction, I get that $f$ is the zero function, if $f$ would be continuous. So, the answer should be true, but how can I prove this fact. Thank you.
This is not necessarily true. For a counterexample, start with any arbitrary discontinuous function $f_0 : (-2,-1] \cup [1,2) \to \mathbb{R}$. Then extend $f_0$ to $f: \mathbb{R} \to \mathbb{R}$ so that $f(x) + f(2x) = 0$ for all $x$.
Explicitly, we can define $f$ from $f_0$ as $$ f(x) := \begin{cases} 0 &\text{if } x = 0 \\ (-1)^k f_0\left(2^k x \right) &\text{if } x \ne 0, \text{ where } k \in \mathbb{Z} \text{ is the unique integer such that } 1 \le |2^k x| < 2.\\ \end{cases} $$