Is f continuous everywhere

(cos y) sin x If x ≠0, let f(x, y) = cos y If x = 0, is f continuous at (0, 0)? Is f continuous everywhere? Justify your answers.

The Correct Answer and Explanation is:

To determine whether the function f(x,y)f(x, y)f(x,y) is continuous at (0,0)(0, 0)(0,0) and whether it is continuous everywhere, let’s analyze the function carefully.

Given that: f(x,y)={(cos⁡y)sin⁡xif x≠0cos⁡yif x=0f(x, y) = \begin{cases} (\cos y) \sin x & \text{if } x \neq 0 \\ \cos y & \text{if } x = 0 \end{cases}f(x,y)={(cosy)sinxcosy​if x=0if x=0​

1. Continuity at (0,0)(0, 0)(0,0):

To check continuity at (0,0)(0, 0)(0,0), we need to verify if the following condition holds: lim⁡(x,y)→(0,0)f(x,y)=f(0,0).\lim_{(x, y) \to (0, 0)} f(x, y) = f(0, 0).(x,y)→(0,0)lim​f(x,y)=f(0,0).

First, calculate f(0,0)f(0, 0)f(0,0):
When x=0x = 0x=0, f(0,0)=cos⁡(0)=1f(0, 0) = \cos(0) = 1f(0,0)=cos(0)=1.

Now, evaluate the limit lim⁡(x,y)→(0,0)f(x,y)\lim_{(x, y) \to (0, 0)} f(x, y)lim(x,y)→(0,0)​f(x,y) for different paths:

• Path 1 (along x=0x = 0x=0):
  Along this path, the function is f(0,y)=cos⁡yf(0, y) = \cos yf(0,y)=cosy. As y→0y \to 0y→0, cos⁡y→1\cos y \to 1cosy→1. Hence, lim⁡(0,y)→(0,0)f(0,y)=1\lim_{(0, y) \to (0, 0)} f(0, y) = 1lim(0,y)→(0,0)​f(0,y)=1.
• Path 2 (along y=0y = 0y=0):
  Along this path, the function is f(x,0)=sin⁡xf(x, 0) = \sin xf(x,0)=sinx. As x→0x \to 0x→0, sin⁡x→0\sin x \to 0sinx→0. Hence, lim⁡(x,0)→(0,0)f(x,0)=0\lim_{(x, 0) \to (0, 0)} f(x, 0) = 0lim(x,0)→(0,0)​f(x,0)=0.

Since the limit depends on the path taken (0 along y=0y = 0y=0 and 1 along x=0x = 0x=0), the limit does not exist at (0,0)(0, 0)(0,0). Therefore, the function is not continuous at (0,0)(0, 0)(0,0).

2. Continuity Everywhere:

Now, we need to check if f(x,y)f(x, y)f(x,y) is continuous everywhere.

• When x≠0x \neq 0x=0, f(x,y)=(cos⁡y)sin⁡xf(x, y) = (\cos y) \sin xf(x,y)=(cosy)sinx, and both cos⁡y\cos ycosy and sin⁡x\sin xsinx are continuous functions. Thus, f(x,y)f(x, y)f(x,y) is continuous when x≠0x \neq 0x=0.
• However, as we saw in the previous part, f(x,y)f(x, y)f(x,y) is not continuous at (0,0)(0, 0)(0,0), meaning that the function is not continuous at all points where x=0x = 0x=0.

Thus, the function is not continuous everywhere.

Conclusion:

• f(x,y)f(x, y)f(x,y) is not continuous at (0,0)(0, 0)(0,0).
• f(x,y)f(x, y)f(x,y) is not continuous everywhere.