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Chapter 12: Problem 49

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$g(x, y)=\sqrt[3]{x^{2}+y^{2}-9}$$

Short Answer

Expert verified
Answer: The function \(g(x, y)\) is continuous at all points in \(\mathbb{R}^{2}\).

Step by step solution

01

Determine the domain of the function

To determine the domain of the function, we must first identify the values of \((x, y)\) for which the function is defined. In this case, the function \(g(x, y)\) would be defined for all values of \((x, y)\) in \(\mathbb{R}^{2}\) since the cube root is defined for any real input.
02

Determine the continuity of the function

Now that the domain of the function is established, let's determine its continuity. Recall that a function is continuous at every point in its domain if the limit exists and is equal to the function's value at that point. Since the cube root function is continuous for all real numbers, we must consider only the function inside the cube root, \(x^{2} + y^{2} - 9\). This is a quadratic function that is continuous for all real values of \((x, y)\). Therefore, the function inside the cube root is continuous everywhere in \(\mathbb{R}^{2}\). As a result, the function \(g(x, y) = \sqrt[3]{x^{2} + y^{2} - 9}\) is continuous everywhere in \(\mathbb{R}^{2}\).
03

Identify the continuous points

Since we determined that the function \(g(x, y)\) is continuous for all real values of \((x, y)\), the answer to the exercise is that the function is continuous at all points in \(\mathbb{R}^{2}\).

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