I thought of this construction: Let f, the first function have period 1, g the second have period √2. Define the equivalence relationship that x~y if x = y +

I thought of this construction: Let f, the first function have period 1, g the second have period √2. Define the equivalence relationship that x~y if x = y + a + b√2 for some a and b. Pick for each class a unique representative using the Axiom of Choice. For each representative e, define f(a+b√2+e) = b√2, g(a+b√2+e) = a + e By construction, any number can be represented through a unique (a, b, e). And f and g manifestly have the right periods.