How do you check the continuity of a multivariable function in a region bounded by two or more curves in the xy plane?

Question

anonymous · Answer

If the limit exists on every point of the function

anonymous · Answer

It's kinda hard to show that a limit exists with multivariables since we can approach any point in the function in infinity ways

anonymous · Answer

Everything that we used in single variables carries over like the squeeze theorem

anonymous · Answer

ok so i want to check the continuity of (x^4+4y^2) in the region bounded by y=x^2 and y=2x, which is continuous obviously, shouldn't there be a more concrete method to do the same for more complex functions

anonymous · Answer

The function is going to be continuous on all points in a polynomial. Unless you have have a function that is devided by the variables like in $$xy ^{2}/ x ^{2} + y ^{4} $$  
in this case the function is not continuous at (0,0) because
$$\lim_{(x,y) \rightarrow (0,0)} x*y ^{2}/x ^{2}+ y ^{4} $$  does not exists

anonymous · Answer

but for functions which are discontinuous at more than one point, or for trigonometric functions with infinite discontinuities, how can you say that a function is continuous in a region or not??