M25008PP7

M25008PP7 - Mathematics 250 Fall 2008 Proof Practice # 7...

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Mathematics 250 Fall 2008 Proof Practice # 7 Given open domains U R m , V R , and functions F : U V and G : V R k , show that if F and G are continuous, then the composition G F is also continuous. Response: Some of you approached this through the ± - δ definition of continuity, and others approached it through the characterization of continuity by means of inverse images of open sets. Either approach is viable. We will discuss both. For the ± - δ definition, given a point p o in U , and ± > 0, we would want to find δ > 0 such that when || x - p o || < δ , we also have || ( G F )( x ) - ( G F )( p o ) || < ± . To do this, we first use the fact that G is continuous. This means that, for the given ± and the point F ( p o ), we can find δ 1 > 0 such that if y is in V and || y - F ( p o ) || < δ 1 , then || G ( y ) - G ( F ( p o )) || < ± . Then, since F is continuous, since δ 1 > 0, we can find δ > 0 such that if || x - p o || < δ , then || F ( x ) - F ( p o ) || < δ
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This note was uploaded on 09/24/2009 for the course MATH 250 taught by Professor Rogerhowe during the Fall '06 term at Yale.

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