Uniform continuity
Continuous functions, open intervals and uniform continuity
Card 1 of 3: Uniform continuity
For a continuous function on a closed interval \( [a, b] \) we know it must be bounded; other nice properties follow. If we relax the restriction and consider a continuous function on an open interval, \( (a, b) \), we cannot imply that it is bounded, and we lose subsequence nice properties. Consider \( f: (0, 2) \to \mathbb{R} \) with \( f(x):=\frac{1}{x} \) for an illustration.
Measure of stability
We try to rebuild some new structure around functions on an open interval. The plan is to classify and distinguish continuous functions based on a measure of stability: measure the degree to which they vary over a certain sized interval of their domain.
\( \delta \)-islands of stability
For a function \( f : X \to \mathbb{R} \) to be continuous at \( x_0 \in X \), for any \( \varepsilon > 0 \) there needed to exist some \( \delta > 0 \) such that for all \(x \in (x_0 - \delta, x_0 + \delta)\) we have \( |f(x)-f(x_0)| < \varepsilon \). The \( \delta \) was free to change for both every \( \varepsilon \) and for every \( x_0 \).
Restricting \( \delta \) to be a single value for any \( x_0 \in X \) we can describe what it means for a function to be uniformly convergent over it's full domain.
Uniform continuity
Let \( X \subseteq \mathbb{R} \) be a set, and let \( f : X \to \mathbb{R} \) be a continuous function. We say that \( f \) is uniformly continuous iff for every real \( \varepsilon > 0 \), there exists a real \( \delta > 0 \) such that \( |f(x_1) - f(x_2)| < \varepsilon \) whenever \( x_1, x_2 \in X \) and \( |x_1 - x_2 | < \delta \).
Tao uses an if rather than iff relationship in his definition, but I don't see why.
Continuity vs. uniform continuity—the difference of \( \delta \)
A function \( f : X \to \mathbb{R} \) converges at \( x_0 \) on \( E \) if for every \( \varepsilon > 0 \) there exists a \( \delta > 0 \) such that for all \( x \in \{ i \in X \cap E : |i - x_0 | < \delta \} \) we have \( |f(x) - L| < \varepsilon \) for some real \( L \). \( f \) is continuous at \( x_0 \) if it converges to \( f(x_0) \) at \( x_0 \) on \( X \).
We can see that every uniformly continuous function is continuous but not the other way around.
Example
The function \( f : (0, 2) \to \mathbb{R} \) defined by \( f(x) := \frac{1}{x} \) is continuous on \( (0, 2) \), but not uniformly continuous; the dependency of \( \varepsilon \) on \( \delta \) prescribed by continuity 'becomes worse' as \( x \to 0 \).
