Continuity in multi-variable calculus

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Where is the function

$${f (x, y) = x^2 + x y + y^2}$$

continuous?

How do I go about solving such problems?
 

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  • #2
LCKurtz
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Presumably you have theorems like sums, products, and quotients where the denominator is nonzero, of continuous functions are continuous. And polynomials are continuous. Given that, since your example is a polynomial, there is nothing to do but quote the theorem. More typically you would look for points where the function fails to be continuous. For example if $$
f(x,y) = \frac{x^2+xy+y^2}{x-y}$$you could invoke that polynomials are continuous, so the quotient of two polynomials is continuous except where the denominator is zero. So this example would be continuous except on the line ##y=x##.
 
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Thanks.
 

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