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Chapter 13: Problem 43

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

Short Answer

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

Step by step solution

01

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: $$\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$$
02

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.
03

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: $$f(x, y) = h(g(x, y))$$
04

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.
05

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 \((x, y) \in \mathbb{R}^{2}\).

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Most popular questions from this chapter

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{\sqrt{x^{2}+y^{2}}}$$

A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(f(x)=x^{2}\) ). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on \(\mathbb{R}^{2}\). a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum.

A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.

Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)} c f(x, y)=c \lim _{(x, y) \rightarrow(a, b)} f(x, y)$$

Identify and briefly describe the surfaces defined by the following equations. $$y^{2}-z^{2}=2$$
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