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 1 . Since F ( x ) is in V and within distance δ 1 of F ( p o ), it follows from our choice of δ 1 that || G ( F ( x )) - G ( F ( p o )) || < . Since ( G F )( x ) = G ( F ( x )) for any point x in U
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