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Unformatted text preview: f ( x,y,z ) is continuous where it is deFned. Proof. Consider the functions on R 2 add ( x 1 ,x 2 ) = x 1 + x 2 sub ( x 1 ,x 2 ) = x 1 x 2 mul ( x 1 ,x 2 ) = x 1 x 2 div ( x 1 ,x 2 ) = x 1 x 2 ; each of the Frst three is continuous on R 2 , and the last is continuous o the x 2axis, because of the basic laws about arithmetic of convergent sequences ( Calculus Deconstructed , Theorem 2.4.1). But then application of Remark 3.1.2 to these and powers, roots, exponentials, logarithms and trigonometric functions (which are all continuous where deFned) yields the lemma....
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 Fall '08
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 Calculus, Continuity, Limits

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