## anonymous 5 years ago How do you check the continuity of a multivariable function in a region bounded by two or more curves in the xy plane?

1. anonymous

If the limit exists on every point of the function

2. anonymous

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

3. anonymous

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

4. anonymous

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

5. anonymous

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

6. anonymous

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??