Working from "Principles of Mathematical Analysis", by Walter Rudin I have gleaned the following definition of continuity of a function (which maps a subset one metric space into another):(adsbygoogle = window.adsbygoogle || []).push({});

Suppose [itex]f:E\rightarrow Y[/itex], where [itex]\left( X, d_{X}\right) \mbox{ and } \left( Y, d_{Y}\right) [/itex] are metric spaces, and [itex]E\subset X[/itex]. Let [itex]x\in E[/itex] be a limit point of E.

f is continuous at x if, and only if, for every sequence [itex]\left\{ x_{n}\right\} \rightarrow x[/itex] such that [itex]x_{n}\in E\forall n\in\mathbb{N}[/itex], we have [itex]f(x_{n})\rightarrow f(x)\mbox{ as }n\rightarrow\infty[/itex].

My question is, when generalizing the above definition to multivariate functions, the last line of the definition would include which of the following:

a. [tex]f(x_{n},y_{n})\rightarrow f(x)\mbox{ as }n\rightarrow\infty[/tex],

or

b. [tex]f(x_{n},y_{m})\rightarrow f(x)\mbox{ as }n,m\rightarrow\infty[/tex] ?

I am uncertian if I need the double limit. :rofl: DUH! I get: X is a metric space, it could be of an arbitrary dimension if need be. But suppose that it were a product of metric spaces (with different metrics,) would the question then merit investigation?

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# Def. Continuity in terms of sequences: How do I generalize to multivariate fcns?

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