Homework Help: Proof function is integrable

1. Apr 19, 2012

flurrie

Hello everyone,
I am given the following:
D=[0,1]×[0,1] and f(x,y)=$\frac{1}{1+x+y}$ on D.
a) why is f integrable on D?

I know that the function is integrable because it is bounded and has a finite amount of points where it is discontinuous.
That f is bounded follows directly from the given information.
I also know where f is discontinuous, obviously in the point (0,0)
And I can determine the upper and lower bounds for y and x.
by looking at the boundaries: $\frac{1}{1+x+y}$=0
and $\frac{1}{1+x+y}$=1

so for y:
y=$\varphi$$_{1}$(x)=0
y=$\varphi$$_{2}$(x)=-x
and for x:
x=$\varphi$$_{1}$(y)=0
x=$\varphi$$_{1}$(y)=-y

So I thought that this mend that f(x,y) is discontinuous at the points: (0,0),y=-x,x=-y
but I am not sure because by writing x and y in terms of the other variable I can draw some graphs within D and there the function for y=-x is discontinuous at the boundary x=1 and x=0 and for x=-y is this function discontinuous at y=1 and y=0.

So I'm having a bit of a problem with understanding when a function is discontinuous when there are 2 variables. And I am also wondering, if I can find the points where f is discontinuous, then is it enough to just state that f is discontinuous at those points or do I have to give some kind of proof?

Last edited: Apr 19, 2012
2. Apr 19, 2012

HallsofIvy

You appear to not understand the basic notation:
"And I can determine the upper and lower bounds for y and x.
by looking at the boundaries: $\frac{1}{1+x+y}= 0$
and $\frac{1}{1+x+y}= 1$"

You are given that "D=[0,1]×[0,1]" which means that the boundaries are the lines x= 0, y between 0 and 1, x= 1, y between 0 and 1, y= 0, x between 0 and 1, and y= 1, x between 0 and 1. The fact that the domainis [0, 1]x[0, 1] says nothing about the value of the function.

Also you say "I also know where f is discontinuous, obviously in the point (0,0)" when, in fact, the function is obviously NOT discontinuous there. How did you arrive at that conclusion?

3. Apr 19, 2012

flurrie

Yes you're right, when I looked at it later on I saw I was wrong. Somehow I got confused with x and y-simple
But when you look at f = $\frac{1}{1+x+y}$ I can't find any point where it is discontinuous within D=[0,1]×[0,1] . it would be discontinuous if x=0,y=-1 or x=-1,y=0 right?
so would that mean that this function has no discontinuous points on the interval D?

Last edited: Apr 19, 2012