Examine the continuity of f : R2 → R at (0; 0), where for all (x; y) ∈ R2,

1.)f(x; y) :=
{xy cos(1/x) if x ̸= 0;
0 if x = 0:}

2.)f(x; y) :=
{1 if x > 0 & 0 < y < x2;
0 otherwise:}

To examine the continuity of a function at a particular point, we need to check three conditions:

1. The function must be defined at that point.
2. The limit of the function as (x, y) approaches the given point must exist.
3. The limit of the function as (x, y) approaches the given point must be equal to the value of the function at that point.

Let's analyze the continuity of the given functions at (0, 0) one by one:

1. For the function f(x, y) := xy * cos(1/x) if x ≠ 0; 0 if x = 0:

a) The function is defined for all (x, y) in R^2. It is defined as xy * cos(1/x) unless x = 0, in which case it is defined as 0. So, the function is defined at (0, 0) as f(0, 0) = 0.

b) To check if the limit of the function exists, we need to evaluate the limit as (x, y) approaches (0, 0). We will consider approaching along different paths.

Approaching along the x-axis (y = 0):
lim(x,0)→(0,0) f(x, 0) = lim(x,0)→0 xy * cos(1/x) = 0
Here, the limit approaches 0 as x approaches 0.

Approaching along the y-axis (x = 0):
lim(0,y)→(0,0) f(0, y) = 0
In this case, the limit is always 0 as (0, y) approaches (0, 0).

Approaching along the line y = mx:
lim(x,mx)→(0,0) f(x, mx) = lim(x,mx)→0 xy * cos(1/x)
For any fixed value of m, as x approaches 0, the term xy approaches 0, and cos(1/x) remains bounded between -1 and 1. Therefore, the limit is 0 as (x, mx) approaches (0, 0).

Since the limit of the function is 0 regardless of the path taken, it exists.

c) Finally, we need to check if the limit is equal to the value of the function at (0, 0). Here, f(0, 0) = 0. Since the limit as (x, y) approaches (0, 0) is equal to the value of the function at (0, 0), we can conclude that the function f(x, y) := xy * cos(1/x) if x ≠ 0; 0 if x = 0 is continuous at (0, 0).

2. For the function f(x, y) := 1 if x > 0 and 0 < y < x^2; 0 otherwise:

a) The function is defined for all (x, y) in R^2 based on the given condition.

b) To check the limit of the function as (x, y) approaches (0, 0), we consider different paths.

Approaching along the x-axis (y = 0):
lim(x,0)→(0,0) f(x, 0) = lim(x,0)→0 0 = 0

Approaching along the y-axis (x = 0):
lim(0,y)→(0,0) f(0, y) = 0

Approaching along the line y = mx:
lim(x,mx)→(0,0) f(x, mx) = lim(x,mx)→0 0 = 0

In all cases, the limit of the function as (x, y) approaches (0, 0) is 0.

c) The value of the function at (0, 0) is f(0, 0) = 0.

Since the limit of the function as (x, y) approaches (0, 0) is equal to the value of the function at (0, 0), we can conclude that the function f(x, y) := 1 if x > 0 and 0 < y < x^2; 0 otherwise is continuous at (0, 0).

In summary:

1. The function f(x, y) := xy * cos(1/x) if x ≠ 0; 0 if x = 0 is continuous at (0, 0).
2. The function f(x, y) := 1 if x > 0 and 0 < y < x^2; 0 otherwise is continuous at (0, 0).