# hw7 - Modern Analysis, Homework 7 Due July 29, 2010 1. [12]...

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Modern Analysis, Homework 7 Due July 29, 2010 1. [12] (5 pts) Rudin, Ch 4, problem 2. 2. [16] (5 pts) Suppose that A and B are connected subsets of a metric space ( X,d ) and that A B 6 = {} . Prove that A B is connected. 3. [13] (5 pts) Let f : X Y be a mapping of metric spaces and p be a point in X . Recall our deﬁnition of the oscillation of f at p : osc p f = inf δ> 0 sup x N δ ( p ) d Y ( f ( x ) ,f ( p )) . Prove that if osc p f = 0, then f is continuous at p . 4. [23] (5 pts) Suppose f : R R is uniformly continuous. Prove that there exist constants A,B > 0 such that for all x R we have | f ( x ) | ≤ A | x | + B. 5. [26] (5 pts) A real function f : ( a,b ) R is called convex if for all x,y ( a,b ) and all λ [0 , 1] we have f ( λx + (1 - λ ) y ) λf ( x ) + (1 - λ ) f ( y ) . Prove that every convex function is continuous. Hint: Draw a picture of the convexity con- dition, and notice it means that f is “below lines”. Given p R , and

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## This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.

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hw7 - Modern Analysis, Homework 7 Due July 29, 2010 1. [12]...

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