Large free linear algebras of real and complex functions

Let X be a set of cardinality κ such that κω=κ. We prove that the linear algebra RX (or CX) contains a free linear algebra with 2κ generators. Using this, we prove several algebrability results for spaces CC and RR. In particular, we show that the set of all perfectly everywhere surjective functions f:C→C is strongly 2c-algebrable. We also show that the set of all functions f:R→R whose sets of continuity points equals some fixed Gδ set G is strongly 2c-algebrable if and only if R⧹G is c-dense in itself.