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Chapter 13: Problem 45

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

Short Answer

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

Step by step solution

01

Identify the Trigonometric Function

We can identify that the given function is a trigonometric function with \(\cos(\cdot)\) as the function. The properties of this type of function will help us determine the continuity of \(h(x, y)\).
02

Analyze Continuity of Cosine

From the properties of trigonometric functions, it can be noted that the cosine function is continuous over its entire domain. In other words, \(\cos(\cdot)\) is continuous everywhere in \(\mathbb{R}\).
03

Check Whether the Sum Affects Continuity

In the given function, \(h(x, y) = \cos(x + y)\). We will now analyze the continuity of the cosine function given the sum of \(x\) and \(y\). Since both \(x\) and \(y\) come from the same domain as the cosine function, which is \(\mathbb{R}\), the sum \((x + y)\) will also be in \(\mathbb{R}\). Thus, the cosine function is also continuous with respect to the argument \((x + y)\).
04

Combine Information and State Final Answer

We have learned that the cosine function is continuous everywhere in its domain and that the sum of the variables \((x + y)\) does not affect the continuity of the function. Therefore, the given function \(h(x, y) = \cos(x+y)\) is continuous everywhere in \(\mathbb{R}^2\). So, the function is continuous at all points of \(\mathbb{R}^2\).

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Most popular questions from this chapter

Let \(x, y,\) and \(z\) be nonnegative numbers with \(x+y+z=200\). a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\). b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\). c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\). d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\).

Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)}(x+y)=a+b . \text { (Hint: Take } \delta=\varepsilon / 2 \text { ) }$$

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Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$f(x, y)=\sin ^{-1}(x-y)^{2}.$$
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