A particularly nasty delta-epsilon

[ssayan3]

Homework Statement

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

Homework Equations

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?

 

[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.