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Engineering Calculus Notes 229

Engineering Calculus Notes 229 - f x,y,z is continuous...

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3.1. CONTINUITY AND LIMITS 217 Remark 3.1.2. Suppose f ( −→ x ) is continuous on D R 3 . 1. If g : R R is continuous on G R and f ( −→ x ) G for every −→ x = ( x,y,z ) D , then the composition g f : R 3 R , defined by ( g f )( −→ x ) = g ( f ( −→ x )) i.e., ( g f ) ( x 1 ,...,x n ) = g ( f ( x,y,z )) is continuous on D . 2. If −→ g : R R 3 is continuous on [ a,b ] and −→ g ( t ) D for every t [ a,b ] , then f −→ g : R R , defined by ( f −→ g )( t ) = f ( −→ g ( t )) i.e., ( f −→ g )( t ) = f ( g 1 ( t ) ,g 2 ( t ) ,g 3 ( t )) is continuous on [ a,b ] Second, functions defined by reasonable formulas are continuous where they are defined: Lemma 3.1.3. If f ( x,y,z ) is defined by a formula composed of arithmetic operations, powers, roots, exponentials, logarithms and trigonometric functions applied to the various components of the input, then
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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 2-axis, 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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