Problem 45 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(h(x, y)\).

Step 2: 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 \(\mathbb{R}\).

Step 3: 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 \((x + y)\).

Step 4: 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 \(\mathbb{R}^2\).