3-Limits_1949838284.pdf - 3.1 Limits Recall Definition A...

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1 3.1 Limits Recall: Definition: A vector-valued function f: n m is a rule or process that assigns to each input x in n corresponding output y in m if m > 1. A real function f of n variables is a rule that assigns to each input x or n-tuple of real numbers (x 1 , x 2 , …, x n ) in a domain D n a unique real number, denoted f (x 1 , x 2 , …, x n ). So a real function f of n variables is the same as a real function of a vector x = [x 1 , x 2 , …, x n ], or a real value function of a point (x 1 , x 2 , …, x n ). Let x 0 be a point in n . Open disk or open ball D r ( x 0 ) = { x n | || x x 0 || < r} n . Definition: Let U n . For each point x 0 in U, there exists some r > 0 such that D r ( x 0 ) U. Then U is called as an open set of n . Theorem: D r ( x 0 ) is an open set.
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