A particularly nasty delta-epsilon

1. Dec 13, 2009

ssayan3

1. The problem statement, all variables and given/known data

Prove the function f(x,y) = x/y is continuous.
As an added stipulation, the quotients of limits theorem may not be used.

2. Relevant equations

3. The attempt at a solution
I have absolutely no idea how to go about this one. I can't even get a start on the scratchwork... Can anyone give me any hints to push me along the right way?

2. Dec 13, 2009

LCKurtz

Here's a start. Pick (x0,y0) with y0 not equal zero.

$$\left |\frac x y - \frac {x_0} {y_0}\right |= \left |\frac{y_0x-x_0y}{yy_0}\right |$$

Now subtract and add y0x0 in the numerator and see if you can make the fraction small. You have to get the numerator small and keep the denominator away from 0.