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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 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 $$x \to 0$$.

p244-246