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Chapter 12: Problem 43

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\sin x y$$

Short Answer

Expert verified
The function \(f(x, y) = \sin(xy)\) is continuous at all points \((x, y) \in \mathbb{R}^2\), as the sine function is smooth and continuous for any value of its argument and the argument \(xy\) is a product of two continuous variables.

Step by step solution

01

Examine the function

Take a look at the function \(f(x, y) = \sin(xy)\). Note that the sine function has the properties of being smooth and continuous for any value of their argument.
02

Determine continuity

Since the sine function is continuous and the only argument it takes in function \(f(x, y)\) is \(xy\), which is a product of two continuous variables, we can conclude that the function \(f(x, y) = \sin(xy)\) remains continuous at all points of \(\mathbb{R}^2\).
03

State the conclusion

The function \(f(x, y) = \sin(xy)\) is continuous at all points \((x, y) \in \mathbb{R}^2\).

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