Finite Set of Continuous Functions between Metric Spaces is Pointwise Equicontinuous

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Theorem

Let $\struct {X, d}$ and $\struct {Y, d'}$ be metric spaces.

Let $\map \CC {X, Y}$ be the set of continuous functions $X \to Y$.

Let $\FF = \set {f_1, f_2, \ldots, f_n}$ be a finite subset of $\map \CC {X, Y}$.


Then $\FF$ is pointwise equicontinuous.


Proof

Let $x \in X$.

Let $\epsilon \in \R_{>0}$.

Let $i$ be a natural number with $1 \le i \le n$.


Since $f_i$ is continuous at $x$, there exists $\delta_i > 0$ such that whenever:

$\map d {x, y} < \delta_i$

we have:

$\map {d'} {\map {f_i} x, \map {f_i} y} < \epsilon$

Let:

$\ds \delta = \min_i \set {\delta_i}$

so that:

$\delta \le \delta_i$

for each $i$.

Then, whenever:

$\map d {x, y} < \delta$

we have:

$\map {d'} {\map {f_i} x, \map {f_i} y} < \epsilon$

for each $i$.

So $\FF$ is pointwise equicontinuous.

$\blacksquare$