Cardinality of set of Baire functions

Question

I'm reading this paper of Sierpinski. At p.260 he says that it is well known that the set of all injective Baire functions (on the reals) is of cardinality $2^{\aleph_0}$, but he gives no reference. Is there any reference for this result? Alternatively can you give me an hint on how to prove it?

Answer 1

• There are $2^{\aleph_0}= \mathfrak{c}$ many real continuous functions (and certainly that many injective ones, say $x \to ax+b$ for $a \neq 0$).
• From a set functions of size $\mathfrak{c}$ we can form at most $\mathfrak{c}$ many pointwise limit functions, as $\mathfrak{c}^{\aleph_0}=\mathfrak{c}$ etc.
• By induction it follows there are at most $\mathfrak{c}$ many Baire functions on $\Bbb R$.
• We already have the lower bound from injective continuous functions.

So Sierpiński's remark is rather trivial and bog standard (maybe less so in his time).