$f,\alpha$ are continuous $\mathbb{R}\to\mathbb{R}$ functions satisfying:
$$f\big(x+\alpha(x)\big)=f(x)$$
If $f$ is non-constant, must $\alpha$ be constant?
My idea was to use the fact that no real function can have a double period, but I don't know how to apply it, or whether this is the right way to go.
Here's the first example that comes to mind: $$ f(x) = \sin(x^2) $$ In this case, we can take $$ \alpha(x) = \sqrt{x^2+2\pi}-x $$ which is continuous but not constant.
For a somewhat more remarkable example, the function $$ f(x) = \cos(x+\frac{1}{100}\sin(x)) $$ Is periodic in both your sense and in the sense of a constant period, since we may select an $\alpha(x)$ that is itself periodic. The analysis of such functions is important for applications such as FM radio.