Let $A$ be the set of continuous functions from the reals to the reals, let $P$ be the set of periodic functions from the reals to the reals, and $F$ be the set of all functions from the reals to the reals. Let $S$ the set of functions which are expressible as the sum of an element from $A$ and an element from $P$. Note that $A$ has the same cardinality as $\mathbb{R}$, but $P$ has the same cardinality as $2^{|\mathbb{R}|}$ which is the cardinality of $F$. So there is no cardinality based argument that there are elements in $F$ which are not in $S$.
However, it seems like there should be elements in $F$ which are not in $S$, and my guess is that there should be some construction possibly using a Hamel basis.
In that context I have three questions:
- Can we show there is an element of $F/S$?
- Can we prove 1 without assuming the Axiom of Choice?
- Do the answers for 1 and 2 change if in $A$ we allow instead of just continuous functions, functions which are continuous except at countably many points?