Proof function with two variables is continuous $f(x,y) = 2xy/(x^2+y^2)$

Question

I have this function:

$f(x,y)= \frac{2xy}{x^2+y^2}$ if $(x,y)$ is not $(0,0)$.

Now I want to prove that this function is continuous. I have tried using the epsilon delta method, but I don't really know if it works with a two variables function. The $|f(x)-f(a)|$ seems to get very complicated, because of the $(x,y)-(a_1,a_2)$. How should I approach this proof?

You're defining your function in $;\Bbb R^2\setminus{(0,0)};$ and here you have the ratio of two polynomials and the denominator polynomial doesn't vanish....thus the function is continuous as the ratio of two continuous functions. — DonAntonio, Feb 19 '21 at 12:59

score 2 · Answer 1 · answered Feb 19 '21 at 13:14

2

Your function is continuous as a composition of continuous functions on $\mathbb{R}^2 \backslash \{(0,0)\}$ as DonAntonio suggested.

Real Analysis: Continuity of a Composition Function

$z(x,y) = \frac{x}{y}$,

$g(x,y) = 2xy$

$h(x,y) = x^2+y^2$

Then $f(x,y) = z(g(x,y),h(x,y))$

answered Feb 19 '21 at 13:14

lpnorm

727

That's a great solution! Thanks – Carm Feb 19 '21 at 13:15

Proof function with two variables is continuous $f(x,y) = 2xy/(x^2+y^2)$

1 Answers1