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Contionuous Functions of Two or More Variables
11.2 CONTINUOUS FUNCTIONS OF TWO OR MORE VARIABLES Two points (x_{x}, y_{t}) and (x_{2},y_{2}) in the hyperreal plane are said to be infinitely close, (x_{1}, y_{1}) ≈ (x_{2}, y_{2}), if both x_{1} ≈ x_{2} and y_{1} ≈ y_{2}. If Δx = x_{2}  x_{1}, Δy = y_{2}  y_{1}, then the distance between (x_{1}, y_{1} and (x_{2}, y_{2}) is LEMMA 1 Two points are infinitely close to each other if and only if the distance between them is infinitesimal. This lemma can be seen from Figure 11.2.1. (An easy proof of the lemma in terms of vectors was given in Section 10.8.) Figure 11.2.1


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