Delta/Epsilon Proof for General Cases

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SUMMARY

The discussion focuses on proving the continuity of the function f(x,y) = xy at the point (x,y) within a domain D ⊆ R. The user expresses difficulty in generalizing the proof beyond specific cases. A suggested approach involves manipulating the expression xy - ab into the form (xy - ay) + (ay - ab) to facilitate the proof. This method highlights the importance of understanding the epsilon-delta definition of continuity in multivariable calculus.

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  • Epsilon-delta definition of continuity
  • Basic multivariable calculus concepts
  • Understanding of limits in two dimensions
  • Familiarity with algebraic manipulation of functions
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Students of calculus, mathematicians, and anyone interested in understanding the continuity of multivariable functions.

ssayan3
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Hey guys, this isn't a math problem for homework or for a course, but simply for my own interest... This is one that I couldn't crack back in Analysis but that I'd like to get some help on resolving for my own peace of mind:

Prove f(x,y) = xy is continuous at (x,y) in a domain D\subset R. I can do this easily with specific cases of (x,y) but I couldn't ever figure out the general case :-(
 
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Try writing xy - ab = (xy - ay) + (ay - ab) and working with that.
 

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