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Math and science::Analysis::Tao::09. Continuous functions on R

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 [...]; 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 [...] 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.