Delta/Epsilon Proof for General Cases

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The discussion focuses on proving the continuity of the function f(x,y) = xy at a point (x,y) within a domain D subset of R. The user expresses difficulty in resolving the general case despite being able to handle specific instances. A suggested approach involves manipulating the expression xy - ab into two parts: (xy - ay) and (ay - ab). This method aims to clarify the continuity proof by examining the behavior of the function as it approaches the point in question. The conversation highlights the challenge of applying delta/epsilon definitions in multivariable calculus.
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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