Making a function continuous

Question

If we had $f(x,y) = \frac{g(x,y)}{h(x,y)}$ which is not defined for $(0,0)$ (as we divide by zero in this case

and I show that the function $L(x) = \lim_{y \to 0} f(x,y)$ is not continuous... what would the method be to "make" $f(x,y)$ continuous for all real numbers by choosing a specific value for $f(0,0)$? Is it even possible?

What I know is that $L(x)$ is not continuous for $x = 0$, as the limit and the function evaluated at 0 give different values.

Answer 1

If I'm interpreting you correctly, it seems that $L(0)$ is defined, but that $$\lim_{x\to 0}L(x)\ne L(0).$$ In other words, $$\lim_{x\to 0}\lim_{y\to 0}f(x,y)\ne\lim_{y\to 0}f(0,y),$$ but if $f$ could be made continuous at $(0,0)$, then those limits would be the same. Hence, $f$ cannot be made continuous at $(0,0).$