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ch4

# Principles of Mathematical Analysis, Third Edition

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Continuity Limits of Functions 4.1 Definition Let X and Y be metric spaces; suppose E X , f maps E into Y , and p is a limit point of E . We write f ( x ) q as x p , or lim x p f ( x ) = q if there is a point q Y with the following property: For every ǫ > 0 there exists a δ > 0 such that d Y ( f ( x ) , q ) < ǫ for all points x E for which 0 < d X ( x, p ) < δ. 4.2 Theorem Let X , Y , E , f , and p be as in Definition 4.1. Then lim x p f ( x ) = q if and only if lim n →∞ f ( p n ) = q for every sequence { p n } in E such that p n negationslash = p, lim n →∞ p n = p. Corollary If f has a limit at p , this limit is unique. 4.4 Theorem Suppose E X , where X is a metric space, p is a limit point of E , f and g are complex functions on E and lim x p f ( x ) = A, lim x p g ( x ) = B. Then we have (a) lim x p ( f + g )( x ) = A + B ; (b) lim x p ( fg )( x ) = AB ; (a) lim x p parenleftbigg f g parenrightbigg ( x ) = A B , if B negationslash = 0 . 1

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Continuous Functions 4.5 Definition Suppose X and Y are metric spaces, E X , p E , and f maps E into Y . Then f is said to be continuous at p if for every ǫ > 0 there exists a δ > 0 such that d Y ( f ( x ) , f ( p )) < ǫ for all points x E for which d X ( x, p ) < δ . If f is continuous at every point of E , the f is said to be continuous on E . It should be noted that f has to be defined at the point p in order to be continuous at p . 4.6 Theorem In the situation given in Definition 4.5, assume also that p is a limit point of E . Then f is continuous at p if and only if f ( x ) f ( p ) as x p . 4.7 Theorem Suppose X , Y , Z are metric spaces, E X , f maps E into Y , g maps the range of f , f ( E ) , into Z , and h = g f (i.e. h : E Z , h ( x ) = g ( f ( x )) for x E ). If f is continuous at a point p E and if g is continuous at the point f ( p ) , then h is continuous at p .
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