Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$

You might be interested in problem 12 in "functional equations" by Marko Radovanović

The problem states: Find all functions $f,g,h\colon \mathbb R\to\mathbb R$ that satisfy $f(x+y)+g(x-y)= 2h(x)+2h(y)$. The solution is quite short and beautiful (as is typical with that kind of problems.)

Turns out the only solutions are $$h(x) = \alpha x^2 + \beta x + b$$ $$f(x) = \alpha x^2 + 2\beta x + 4 b- a$$ $$g(x) = \alpha x^2 + a$$

For some choice of constants. Specializing to the case where $f=g=h$ yields $\beta = 0$ and $a=b=4b-a$, so $f(x)=g(x)=h(x)=\alpha x^2$.

Edit: I just noted that the author apparently reduces this problem to something very similar to the given problem (the sign on the left hand side is different) and then states it is trivial with the given technique. I think it's not that difficult, something like this might work on his problem --- maybe it provides a shortcut here too (?):

• Show that $f(0) = 0$ and $f(-x) = f(x)$
• Assume that $f(1)=f(-1) = \alpha$, find $f(2)=f(1+1)=4\alpha$
• Find that $f(n) = f((n-1)+1) = n^2 \alpha$ with induction.
• Proceed to deduce that $f(\tfrac{n}{m}) = \frac{n^2 \alpha}{m^2}$.
• Use continuity.