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....
View
Full Document
 Fall '08
 ALL
 Calculus, Continuity, Derivative, Limits, Inverse function, Exponentials, continuous oﬀ

Click to edit the document details