# Thread: Determining continuity of a two variable equation?

1. ## Determining continuity of a two variable equation?

Here is the question...
Determine the points (x,y), if any, at which f(x,y) is not continuous.
a.$\displaystyle f(x,y) = \frac{x-y}{1+x^2+y^2}$

b.$\displaystyle f(x,y)=e^{x+y}+\sqrt{x+y}$

Can someone please solve these and explain to me how to go about solving them? Thanks in advance.

2. Originally Posted by Infernorage
Here is the question...
Determine the points (x,y), if any, at which f(x,y) is not continuous.
a.$\displaystyle f(x,y) = \frac{x-y}{1+x^2+y^2}$

b.$\displaystyle f(x,y)=e^{x+y}+\sqrt{x+y}$

Can someone please solve these and explain to me how to go about solving them? Thanks in advance.
Are you assuming real values for $\displaystyle x$ and $\displaystyle y$?

If so, the first will be continuous everywhere.

There can only possibly be a discontinuity where the denominator is 0.

But $\displaystyle 1 + x^2 + y^2 > 0$ for all real $\displaystyle x$ and $\displaystyle y$.

3. Originally Posted by Prove It
Are you assuming real values for $\displaystyle x$ and $\displaystyle y$?

If so, the first will be continuous everywhere.

There can only possibly be a discontinuity where the denominator is 0.

But $\displaystyle 1 + x^2 + y^2 > 0$ for all real $\displaystyle x$ and $\displaystyle y$.
Oh okay, that makes sense. What about the second one though, because if x+y is negative than the square root can't be taken, so I'm not sure what to do with that?

4. $\displaystyle f(x,y)=e^{x+y}+\sqrt{x+y}$
x+y is continuous everywhere and so is $\displaystyle e^x$ so $\displaystyle e^{x+y}$ is continous everywhere. $\displaystyle \sqrt{x}$ is continuous for all positive x (and "continuous from the right" at x= 0) so $\displaystyle \sqrt{x}$ is continuous for all (x,y) such that x+y> 0. x+ y= 0 or y= -x is a line through (0,0) and (1,0) has 1+ 0> 0 so $\displaystyle \sqrt{x+y}$ is continuous "above and to the right" of that line.

$\displaystyle f(x,y)= e^{x+ y}+ \sqrt{x+y}$ is continous for (x,y) "above and to the right" of the line y= -x and "continuous from above and to the right:" on that line.