We can say that as the , or .
If, intuitively speaking, by going close enough to (a,b) we can get f(a,b) as close as we like to L.
f(x,y) is said to be continuous at (a,b) if .
Just as with one variable limits from below and above may be different, with two variables we may get various limits coming in from different paths.
Consider coming in to (0,0) along the line y=mx (m fixed). Along this line we have:
Thus along y=mx,
If we come in along the line ,