Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give definitions of continuity at a point and continuity on a set for a function of three variables.

Step-by-Step Solution

Answer

Continuity at a point ensures function values match as variables approach the point; continuity on a set means this condition holds for every point in the set.

Step 1: Understand Continuity at a Point

Continuity at a point for a function of three variables, let's say \( f(x, y, z) \), means that for a given point \( (a, b, c) \), the function approaches the value \( f(a, b, c) \) as the input \( (x, y, z) \) gets arbitrarily close to \( (a, b, c) \). Formally, \( f \) is continuous at \( (a, b, c) \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( \sqrt{(x-a)^2 + (y-b)^2 + (z-c)^2} < \delta \), we have \( |f(x, y, z) - f(a, b, c)| < \epsilon \).

Step 2: Grasp Continuity on a Set

For continuity on a set, consider a function \( f(x, y, z) \) and a set \( S \) within its domain. \( f \) is continuous on \( S \) if it is continuous at every point within \( S \). This means for every point \( (x_0, y_0, z_0) \) in \( S \), the condition for continuity at a point must hold true. Thus, for every \( \epsilon > 0 \), a corresponding \( \delta > 0 \) must exist for all points in \( S \) so that \( |f(x, y, z) - f(x_0, y_0, z_0)| < \epsilon \) whenever \( \sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2} < \delta \).

Key Concepts

Continuity at a Point

When discussing continuity at a point in multivariable calculus, we focus on functions with multiple variables—even three in this case, such as a function noted as \( f(x, y, z) \). Continuity at a point essentially ensures that the function behaves well around that particular point. Consider a specific point \((a, b, c)\). Continuity implies that as we move closer to \((a, b, c)\), the function value \(f(x, y, z)\) should approach \(f(a, b, c)\).

In practical terms, this means no sudden jumps or surprises when you check the function near the point. Formally, we define this using a mathematical concept where the change around \((a, b, c)\) should be kept within desired limits. For every arbitrarily small value \(\epsilon > 0\), there exists another small value \(\delta > 0\) such that:

\[ |f(x, y, z) - f(a, b, c)| < \epsilon \]

whenever:

\[ \sqrt{(x-a)^2 + (y-b)^2 + (z-c)^2} < \delta \].

This means the distance in the function value is small whenever we are close to \((a, b, c)\) in the input space.

Continuity on a Set

Continuity on a set extends the idea of continuity at a single point to an entire collection of points. It means that the function is continuous not just at one point, but at every point within a given set \( S \). Imagine you have an area or region in space and you want the function \( f(x, y, z) \) to be reliably smooth across it.

For the function to be continuous on the set \( S \), it must meet the continuity condition at each point \((x_0, y_0, z_0)\) in \( S \). This means:

• For any given \(\epsilon > 0\), you can find a corresponding \(\delta > 0\).
• The condition \(|f(x, y, z) - f(x_0, y_0, z_0)| < \epsilon\)
• Must hold true whenever \(\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2} < \delta\).

So, regardless of which point you pick in \( S \), the function behaves without abrupt changes across the whole set.

Three Variables Function

In multivariable calculus, a function of three variables, represented as \( f(x, y, z) \), deals with three independent parameters instead of just one or two as you might find in simpler math. Each variable can represent a dimension - think of \(x\), \(y\), and \(z\) as the width, height, and depth in a three-dimensional space.

These types of functions expand our ability to represent and analyze real-world phenomena which involve multiple changing factors. Examples include calculating temperature variation in a room or modeling pressure changes in a hydraulic system.

Understanding a function of three variables means grasping how altering each variable influences the outcome. This makes visualization and conceptualization crucial, as the input changes across a three-dimensional grid or space. Evaluating continuity for such functions helps ensure they behave predictably in practical scenarios, like smoothly transitioning between temperatures or pressures.

By understanding the behavior of a function in three dimensions, we can apply mathematical theory to complex systems and predict how they react to change.