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Chapter 13: Problem 68

State the definition of continuity of a function of two variables.

Short Answer

Expert verified
A function of two variables \( f(x, y) \) is continuous at a point \( (a, b) \) if for every \( \epsilon > 0 \) there is a \( \delta > 0 \) such that if the distance between \( (x, y) \) and \( (a, b) \) is less than \( \delta \), then the absolute value of the difference between \( f(x, y) \) and \( f(a, b) \) is less than \( \epsilon \).

Step by step solution

01

Understanding Continuity

In order to understand the concept of continuity for a function of two variables, recall that we say a function \( f(x, y) \) is continuous at a point \( (a, b) \) if the output of the function approaches a particular finite number as \( (x, y) \) gets arbitrarily close to \( (a, b) \) from any direction.
02

Formal Definition

To put it formally, we say that a function \( f(x, y) \) is continuous at a point \( (a, b) \) if for every positive number \( \epsilon \) there is a positive number \( \delta \) such that if the distance between \( (x, y) \) and \( (a, b) \) is less than \( \delta \) (but not equal to zero), then the absolute value of the difference between \( f(x, y) \) and \( f(a, b) \) is less than \( \epsilon \). This is formally written as: for every \( \epsilon > 0 \) there exists a \( \delta > 0 \) such that if \( 0 < \sqrt {(x-a)^2 + (y-b)^2} < \delta \), then \( |f(x, y) - f(a, b)| < \epsilon \).
03

Interpretation

The interpretation of this definition is that as \( (x, y) \) approaches \( (a, b) \), the value of \( f(x, y) \) should get arbitrarily close to \( f(a, b) \). In other words, the function should not experience any abrupt changes in value - it should change smoothly without any jumps, breaks, or holes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definition of Continuity
Continuity is a fundamental concept in calculus, and it has a particularly interesting application in the realm of multivariable functions. Imagine taking a hike on a smoothly curving hill; you can stroll the entire path without encountering any sharp corners or discontinuities. Similarly, in mathematics, a function of two variables is considered to be continuous if at every point along the surface that it represents, the landscape of values changes in a smooth, uninterrupted manner. For students beginning to explore multivariable calculus, it is key to recognize that continuity ensures a function's behavior is predictable in the immediate vicinity of any point.
Function of Two Variables
Functions can be thought of as machines that take inputs and produce outputs. In multivariable calculus, we frequently encounter functions that require two inputs to determine the output. These are aptly called functions of two variables, and they're written as \( f(x, y) \). Imagine plotting points on a floor where the horizontal dimensions are determined by \( x \) and \( y \), while the vertical dimension represents the function's output. This conceptualization helps to visualize how a function behaves over a whole region, not just at a single point, allowing us to explore the intricate behavior of surfaces and curves in three-dimensional space. Grasping this visualization will aid students in understanding the topology of the function's graph and how it responds to changes in the input values.
Epsilon-Delta Definition
The epsilon-delta definition of continuity is like a mathematical translation of the phrase 'close but not too close.' Here, epsilon (\( \epsilon \)) and delta (\( \delta \)) are used to describe how the outputs and inputs of a function behave when they are within certain bounds of each other. In simple terms, for every tiny bit that we want the output of the function to vary (that's our epsilon), there's a corresponding tiny range for our input variables (that's our delta) that guarantees this condition.

By using this formal definition, we ensure the function doesn't suddenly leap to a different value as we move infinitesimally close to the point of interest. If a function of two variables meets these criteria for all points in its domain, we can rightfully say it's continuous. Students must appreciate that this framework lays the foundations for understanding limits and guarantees functions behave 'nicely'鈥攚ithout any unexpected jumps鈥攁round a given point.

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Most popular questions from this chapter

Use Lagrange multipliers to find the indicated extrema, assuming that \(x, y\), and \(z\) are positive. Maximize \(f(x, y, z)=x y z\) Constraint: \(x+y+z-6=0\)

Prove that \(\lim _{y y \rightarrow(a, b)}[f(x, y)+g(x, y)]=L_{1}+L_{2}\) where \(f(x, y)\) approaches \(L_{1}\) and \(g(x, y)\) approaches \(L_{2}\) as \((x, y) \rightarrow(a, b) .\)

The utility function \(U=f(x, y)\) is a measure of the utility (or satisfaction) derived by a person from the consumption of two products \(x\) and \(y .\) Suppose the utility function is \(U=-5 x^{2}+x y-3 y^{2}\) (a) Determine the marginal utility of product \(x\). (b) Determine the marginal utility of product \(y\). (c) When \(x=2\) and \(y=3\), should a person consume one more unit of product \(x\) or one more unit of product \(y\) ? Explain your reasoning. (d) Use a computer algebra system to graph the function. Interpret the marginal utilities of products \(x\) and \(y\) graphically.

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A cargo container (in the shape of a rectangular solid) must have a volume of 480 cubic feet. The bottom will cost \(\$ 5\) per square foot to construct and the sides and the top will cost \(\$ 3\) per square foot to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has minimum cost.
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