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Intro to Analysis in-class_Part_28

# Intro to Analysis in-class_Part_28 - 4.1 Concepts of...

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Unformatted text preview: 4.1 Concepts of continuity 61 have | f ( x )- f ( t ) | = | 1- | = 1 > 1 2 = ε, so f ( t ) 9 f ( x ). MORAL: the defn of continuity prevents a function from changing too rapidly; | f ( x ) | cannot “jump”. Definition 4.1.7. f is a continuous function iff it is continuous at each x in its domain, i.e., ∀ ε > , ∀ x, ∃ δ, | x- t | < δ = ⇒ | f ( x )- f ( t ) | < ε. Strengthen this idea by disallowing a function from growing faster than a “globally controlled” rate. Definition 4.1.8. f is uniformly continuous on D iff ∀ ε > , ∃ δ, ∀ x, | x- t | < δ = ⇒ | f ( x )- f ( t ) | < ε. This δ depends only on ε , not on x ; thus, it works for ALL x simultaneously. NOTE: uniformly continuous is a global property; it makes no sense to ask if f is uniformly continuous at x . Example 4.1.4. f ( x ) = x 2 is not uniformly continuous on R . ¬ ( ∀ ε > , ∃ δ, ∀ x, ( | x- t | < δ = ⇒ | f ( x )- f ( t ) | < ε )) ∃ ε > , ∀ δ, ∃ x, ¬ ( | x- t | < δ = ⇒ |...
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Intro to Analysis in-class_Part_28 - 4.1 Concepts of...

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