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.