# How to define the domain of a 2-variable function as this?

1. Apr 8, 2012

### loveinla

Hi,

I am wondering how to express the domain of a two-variable function f(x,y) as below.

For any given y, f(x,y) is defined on [0, g(y)); for any given x, f(x,y) is defined on [0, h(x)). Then how should I specify the domain for the function f(x,y)?

To be more specific, f(x,y)=x+y, and for any given y, it is defined on [0, y); for any given x, it is defined on [0, x). I want to know what is the domain of f(x,y).

Thanks.

2. Apr 8, 2012

### conquest

the domain D is in general

$D=\{(x,y)\in R^2| x \in[0,g(y))$ and $y \in [0,h(x))\}$

I don't think that without knowing more about h and g you can say more than this which is just a tidier formulation of what you say in your post.

and in the specific case:

$D=\{(0,0)\}$

maybe even this is excluded since does [0,0) include 0 or not...

Anyway the example says in fact that f(x,y) is only defined for x and y greater or equal zero and x<y and y<x (with the exception above depending on [0,0) including 0 or not).

although the function x+y is then quite arbitrarily restricted since we can easily define that on all of the real plane

Actually now that I think about it the statement is rather contradictory since you say for any given y the function is defined on [0,y) so for instance f(3,4) should then be defined. but you say for any given x it is defined on [0,x) so f(3,4) should not be defined. unless you mean something like at least defined for this and that in which case the formulation of the problem is a bit weird and the answer would simply be:
D is the real plane

and in general

$D=\{(x,y)\in R^2| x \in[0,g(y))$ or $y \in [0,h(x))\}$

I hope this helps!

3. Apr 8, 2012