Continuous Real Function is Baire Function

Theorem

Let $X \subseteq \R$.

Let $f : X \to \R$ be a continuous function.

Then $f$ is a Baire function.

Proof

For each natural number $n$, define:

$\map {f_n} x = \map f x$

Since $f$ is continuous:

$f_n$ is continuous for each $n$.

Clearly, for each $x \in X$ we have:

$\ds \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$

from Eventually Constant Sequence Converges to Constant.

So:

$\sequence {f_n}$ is a sequence of continuous functions that converges pointwise to $f$.

So $f$ is a Baire function.

$\blacksquare$