I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be represented as the sum of two functions that are each symmetric about (possibly distinct) vertical axes of symmetry?
This seems false to me, but it is harder than I expected to find a counterexample. For example, when trying to decompose f(x)=x this way, one can easily see that if our functions are a(x) and b(x), and we can translate and stretch so that we need to solve the functional equations:
but I do not see an obvious contradiction.
Is such a decomposition always possible?