Functions of two variables, continuity

1. Apr 14, 2010

jbarbarez

Consider the following functions each of which is defined on the x - y plane

f1(x) = (x-y)/(x+y) if x + y is not 0 and otherwise f1(x,y) = 0

f2(x,y) = (xy)/(x^2 + y^2) if (x,y) is not (0,0) and otherwise f2(0,0) = 0

f3(x,y) = (x^3 - y^3)/(x^2 + y^2) if (x,y) is not (0,0), and otherwise f3(0,0) is 0

Which of these is continuous

A) none B) f1 only C) f2 only D) f3 only E) all three

I know the defination of continuity for a single variable is the lim as x-> a of f(x) = f(a)

So i assume for two variables it should be the lim as (x,y) -> (a,b) of f(x,y) = f(a,b)

But I am not sure how to figure this out can someone help out please

2. Apr 15, 2010

ystael

You have the correct definition: if $$U$$ is an open set in $$\mathbb{R}^2$$ and $$(a, b) \in U$$, then $$f: U \to \mathbb{R}$$ is continuous at $$(a, b)$$ if $$\lim_{(x, y) \to (a, b)} f(x, y)$$ exists and equals $$f(a, b)$$.

A good way to detect that a function of two real variables is not continuous at a point is to find the limit as you approach the point along lines of different slopes. Often the behavior of the function will be different along different lines.

To prove that a function is continuous at a point, if you don't have another handle on it (such as, the function is built using arithmetic operations and composition on functions that are known to be continuous), start with the epsilon-delta definition of limit.