Definitions: $C(S) = \{f: f \text{ is bounded, continuous and real-valued function on }S\}$
with metric $$d(f,g) = \sup\{|f(x) - g(x)| : x\in S \}$$
Problem:
Consider $S \subseteq \mathbb{R}$. For a fixed $s_0 \in S$, define $F(f) = f(s_0)$. Show $F$ is a uniformly continuous real-valued function on the metric space $C(S)$.
Therefore, we need to show that
$$\forall \epsilon>0, \ \exists \delta > 0 : f,g \in C(S) \ \land \ d(f, g) < \delta \Rightarrow \ |F(f) - F(g)| < \epsilon$$
From the definition of the metric
$$|F(f) - F(g)| = |f(s_0)-g(s_0)| \leq \sup\{|f(x) - g(x)| : x\in S\} = d(f,g) < \delta$$
Let
$$\epsilon = \delta$$
Therefore, $$\text{if } d(f,g) < \delta \Rightarrow |F(f) - F(g)| < \delta = \epsilon$$