Answer:
a) To determine if f(x, y) is continuous at (0, 0), we need to check if the limit of f(x, y) as (x, y) approaches (0, 0) exists and is equal to f(0, 0).
Let's evaluate the limit of f(x, y) as (x, y) approaches (0, 0):
lim(x,y)→(0,0) (1 - xy) = 1 - 0 = 1
Since the limit is equal to f(0, 0) = 1, we can conclude that f(x, y) is continuous at (0, 0).
b) To evaluate fy (0, 0) and fy (0, 0), we need to find the partial derivatives of f(x, y) with respect to y and x, respectively.
Partial derivative fy (0, 0):
fy (x, y) = -x
fy (0, 0) = -(0) = 0
Partial derivative fx (0, 0):
fx (x, y) = -y
fx (0, 0) = -(0) = 0
Therefore, fy (0, 0) = 0 and fx (0, 0) = 0.
c) To determine if f(x, y) is differentiable at (0, 0), we need to check if both partial derivatives fy and fx are defined and continuous at (0, 0).
Since fy (0, 0) = 0 and fx (0, 0) = 0, both partial derivatives exist at (0, 0).
d) To determine if fx and fy are continuous at (0, 0), we need to check if they are continuous in a neighborhood around (0, 0).
In this case, both fx and fy are constants, and constants are continuous everywhere. Therefore, fx and fy are continuous at (0, 0).
e) The theorems mentioned are consistent with the results found in parts a through d.
The first theorem states that if the partial derivatives fx and fy exist and are continuous at a point (a, b), then the function f is differentiable at that point. In our case, both fx and fy exist and are continuous at (0, 0), so f is differentiable at (0, 0).
The second theorem states that if a function is differentiable at a point (a, b), then it is also continuous at that point. Since f is differentiable at (0, 0) according to the first theorem, it is also continuous at (0, 0).
In our analysis, we found that f(x, y) is continuous at (0, 0), both partial derivatives exist and are continuous at (0, 0), and therefore, f is differentiable at (0, 0). This aligns with the theorems mentioned.