M25008PP8 - Mathematics 250 Fall 2008 Proof Practice 8 Let...

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Mathematics 250 Fall 2008 Proof Practice # 8 Let U be an open domain in R m . Let f : U R k and g : U R be functions. Define f g to be the function from U to R k + whose first k coordinates are the coordinates of f , and whose last coordinates are the coordinates of g . In particular, if f and g are scalar-valued, i.e., k = = 1, then ( f g )( u ) = f ( u ) g ( u ) maps U to R 2 . a) Show that if f and g are continuous, then f g is continuous. b) Combine Proof Practices # 2 and # 7 with part a) to show that, if f and g are continuous scalar- valued functions on U , then f + g and fg are also continuous. (This of course can be shown directly. However, you are asked to base your argument on the indicated results.) Response: a) To show that f g is continuous, one can appeal to the result, proved in class, that a vector-valued function F = F 1 F 2 F 3 . . . F n : U R n is continuous if and only if each component F i of F is continuous. By definition of f g
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