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Unformatted text preview: coordinatewise Using this notion, we can deFne continuity of a real-valued function of three (or two) variables f ( x ) by analogy to the deFnition for real-valued functions f ( x ) of one variable: Defnition 3.1.1. A real-valued function f ( x ) is continuous on a subset D R 3 of its domain if whenever the inputs converge in D (as points in R 3 ) the corresponding outputs also converge (as numbers): x k x f ( x k ) f ( x ) . It is easy, using this deFnition and basic properties of convergence for sequences of numbers, to verify the following analogues of properties of continuous functions of one variable. irst, the composition of continuous functions is continuous (Exercise 3 ): 1 When the domain is an explicit subset D R n we will write f : D R ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08