> [!theorem] > > Let $f: X \to Y$ be a [[Continuity|continuous]] function between [[Metric Space|metric spaces]]. $f$ is **uniformly** continuous if > $ > \forall \varepsilon > 0,\exists \delta > 0: \forall c \in D, |x - c| < \delta \Rightarrow|f(x) - f(c)| < \varepsilon > $ > the $\delta$ parameter does not depend on $c$. > > The space $UC(X, Y)$ is the space of all uniformly continuous functions from $X$ to $Y$. > [!theorem] > > Let $X, Y$ be metric spaces. Define a metric on $C(X, Y)$ by > $ > d(f, g) = \sup_{x \in X}d(f(x), g(x)) > $ > The space $UC(X, Y)$ is a closed subset of $C(X, Y)$. > > *Proof*. Let $\seq{f_n}$ be a sequence in $UC(X, Y)$ such that $f_n \to f \in C(X, Y)$. Let $\eps > 0$, then there exists $n \in \nat$ such that $d(f_n, f) < \eps/3$. Since $f_n \in UC(X, Y)$, there exists $\delta > 0$ such that $d(f_n(x), f_n(y)) < \eps/3$ for all $x, y$ with $d(x, y) < \delta$. In which case > $ > \begin{align*} > d(f(x), f(y)) &\le d(f(x), f_n(x)) + d(f_n(x), f_n(y)) + d(f_n(y), f(y)) \\ > &< \varepsilon > \end{align*} > $ > for any $x, y$ with $d(x, y) < \delta$.