Question 6.T.14: If D is a compact subset of R and the function f : D → R is ...
If D is a compact subset of \mathbb{R} and the function f : D → \mathbb{R} is continuous, then f is uniformly continuous.
Let ε > 0 be given. Since f is continuous at every x ∈ D, there is a δ(x) > 0 such that
t ∈ D, |t − x| < δ(x) ⇒ |f(t) − f(x)| < ε. (6.22)
The collection of open intervals
\left\{\left(x − \frac{1}{2}δ(x), x +\frac{1}{2}δ(x) \right) : x ∈ D \right\}
clearly covers D. Since D is compact, there is a finite number of points x_{1}, x_{2}, · · · , x_{n} in D such that
D ⊆ \overset{n}{\underset{i=1}{\cup }} \left(x_{i} − \frac{1}{2}δ(x_{i}), x_{i} +\frac{1}{2}δ(x_{i}) \right). (6.23)
Define
δ = \frac{1}{2} \min \left\{δ(x_{1}), · · · , δ(x_{n})\right\},
which is positive, being the minimum of a finite set of positive numbers.
Now suppose that x, t ∈ D and |t − x| < δ. From (6.23) we conclude that
x ∈ \left(x_{i} − \frac{1}{2}δ(x_{i}), x_{i} +\frac{1}{2}δ(x_{i}) \right) for some i ∈ {1, · · · , n}
⇒ |t − x_{i}| ≤ |t − x| + |x − x_{i}| < δ + \frac{1}{2} δ(x_{i}) ≤ δ(x_{i}).
And from (6.22) we arrive at
|f(t) − f(x)| ≤ |f(x) − f(x_{i})| + |f(x_{i}) − f(t)| < 2ε.