Unformatted text preview: coordinatewise Using this notion, we can deFne continuity of a realvalued function of three (or two) variables f ( −→ x ) by analogy to the deFnition for realvalued functions f ( x ) of one variable: Defnition 3.1.1. A realvalued 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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 Fall '08
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 Calculus, Derivative, Continuous function

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