Problem 43 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall the definition of continuity

A function is continuous at a point \((a, b)\) if the limit as \((x, y) \to (a, b)\) of the function exists and equals the function's value at that point. More formally, \(f(x, y)\) is continuous at \((a, b)\) if: $$\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$$

Step 2: Identify properties of continuous functions

One property of continuous functions is that basic operations (addition, subtraction, multiplication, and division) between continuous functions will give another continuous function. Additionally, if \(g(x)\) and \(h(x)\) are continuous functions, then their composition \(\displaystyle g(h(x))\) is also continuous.

Step 3: Analyze the given function

The given function is \(f(x, y) = \sin x y\). Notice that it is composed of three simpler functions: \(g(x, y) = x y\), \(h(x) = \sin x\), and the identity function \(i(y) = y\). We can rewrite our function as: $$f(x, y) = h(g(x, y))$$

Step 4: Determine continuity of the component functions

Now, we need to find the continuity of the component functions. 1. The functions \(x\) and \(y\) are linear, so they are continuous on \(\mathbb{R}\). 2. The sine function, \(\sin x\), is a trigonometric function, and it is continuous on \(\mathbb{R}\). Since \(g(x, y)\) is a product of continuous functions (i.e. \(x\) and \(y\)) and follows the properties of continuous functions, it is continuous on \(\mathbb{R}^2\). Similarly, as \(h(x) = \sin x\) is continuous on \(\mathbb{R}\), the composition \(\displaystyle f(x, y) = h(g(x, y))\) is also continuous.

Step 5: Produce the final answer

We have shown that the component functions of \(f(x, y) = \sin x y\) are continuous and their composition is also continuous. Therefore, the function \(f(x, y)\) is continuous for all points \((x, y) \in \mathbb{R}^{2}\).