differential calculus – Dirichlet function – i dont have a lead

i have no idea what to do,
i thought mybe there is any connection to that “one” in the numerator but i’m literelly stack over flow and have no idea what can i do to solve it:

$$ let f(x):(-2,2)→(0,∞) be a Continuous function for evry x $$
$$proove that f(x)=D(x) has a solution in the section (-2,2) $$

$$f(x+sqrt{2})=frac{1}{f(x)}$$
$$D(x)= Dirichlet function$$

on the one hand i can say that $f(x)$ is continous, the “image” can get any value and in particular for $ non Q$ numbers → Dirichlet function get value of 1 and i finish. i have a feeling this is not the way to solve it and there is a meaning to the limits they gave in the question.
im incompetent if anyone can help me i will appreceate it a lot !