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Intro to Analysis in-class_Part_29

Intro to Analysis in-class_Part_29 - 4.1 Concepts of...

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4.1 Concepts of continuity 63 x f - 1 ( V ) as soon as x ( p - δ, p + δ ). But x f - 1 ( V ) = f ( x ) V = ⇒ | f ( x ) - f ( p ) | < ε. Corollary 4.1.12. f is continuous iff the preimage of every closed set is closed. Proof. HW. Connectedness Definition 4.1.13. A set X is connected iff it CANNOT be written X = A B, A B = , A, B 6 = , for two open sets A, B . Theorem 4.1.14. The continuous image of a connected set is connected. Proof. Let f : X R be continuous, and let X be connected. Must show f ( X ) is connected. Suppose, by way of contradiction, that f ( X ) = A B is a separation of f ( X ). Then f - 1 ( A ) and f - 1 ( B ) are disjoint nonempty open sets whose union is X . < . A B = = f - 1 ( A B ) = , because x f - 1 ( A B ) = x f - 1 ( A ) and x f - 1 ( B ) = f ( x ) A B. 4.1.4 Related definitions Definition 4.1.15. f is a Lipschitz function (or strongly continuous function) iff | f ( x ) - f ( y ) | ≤ M | x - y | for some constant M . (The Lipschitz constant .) Then | x - y | < ε M = ⇒ | f ( x ) - f ( y ) | < ε.
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64 Math 413 Continuous functions
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