Continuous Real Function is Baire Function
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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.
$\blacksquare$