# Baire functions are constant?

Is it true that every Baire function $f:\mathbb{R}\to \mathbb{N}$ must be constant?

$f:X \to Y$ is a Baire function for $X,Y$ metrizable spaces if $f$ is a member of $F(X,Y)$, where $F$ is the smallest class of function from $X$ to $Y$ containing continuous functions and being closed under pointwise limits.

• How do you define a Baire function from $\Bbb R$ to $\Bbb N$? Have you tried writing down the definitions in full, and see if the answer is positive? – Asaf Karagila Aug 16 '13 at 4:18

Recall that the continuous image of a connected space is connected. Therefore every continuous function from $\Bbb R$ to $\Bbb N$ is constant. It suffices to show, if so, that pointwise limit of constant functions is constant.