Continuity of multivariable functions

[mreaume]

Homework Statement

A function f is defined on the whole of the xy-plane as follows:

f(x,y) = 0 if x=0
f(x,y) = 0 if y = 0
f(x,y) = g(x,y)/(x^2 + y^2) otherwise

a) g(x,y) = 5x^3sin(y)
b) g(x,y) = 6x^3 + y^3
c) g(x,y) = 8xy

For each of the following functions g determine if the corresponding function f is continuous on the whole plane

Homework Equations

A function's limit exists if and only if it is not dependent of the path taken.

The Attempt at a Solution

Since the functions are continuous for all values of x and y, the only restriction on the xy plane is at the point (0,0). So I am trying to find the limit of these functions as (x,y) approaches (0,0).

I have done so for c) using the line x=0 and y=x. These produce two different answers. Therefore, the limit of c does not exist at (0,0) and the function is not continuous on the xy plane.

Any tips as to how I should tackle the other ones? I suspect that their limits are 0 (since every path I try gives 0), but I am having a hard time proving this.

 

[UltrafastPED]

Aren't they continuous at any point where the partial derivatives exist?

 

[mreaume]

Thanks. Turns out I misread the question. I ended up using your tip (partial derivatives) and was able to solve the problem.