1. The problem statement, all variables and given/known data 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 2. Relevant equations A function's limit exists if and only if it is not dependent of the path taken. 3. 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.