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.