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The epsilon-delta definition is a formal way of defining the continuity of a function. It's a rigorous approach used in calculus to ensure that a function behaves predictably in response to small changes. In essence, this definition helps us understand how sensitive a function is to changes in its input near a specific point.
For a function of several variables, the epsilon-delta definition asserts that a function \( f \) is continuous at a point \( (a, b, c) \) if, for every small positive number \( \epsilon \), there exists another small number \( \delta > 0 \) such that whenever the distance from \( (x, y, z) \) to \( (a, b, c) \) is less than \( \delta \), the difference in function values \(|f(x, y, z) - f(a, b, c)|\) is less than \( \epsilon \).

• You choose \( \epsilon \), a measure of how close you want the function's values to be near a point.
• Then you find \( \delta \), which defines how close the input values need to be to ensure the function value stays within \( \epsilon \).

This definition is foundational in proving the continuity of multivariable functions, as demonstrated in the given exercise.

Functions of several variables are equations that depend on more than one input value. Unlike single-variable functions, which involve a line or a curve on a plane, multivariable functions describe surfaces or volumes in higher dimensions.
In multivariable calculus, these functions typically take the form \( f(x, y, z) = x^2 + y^2 + z^2 \), where \( x, y, \) and \( z \) are independent variables. Such functions can have interesting properties and are often explored to understand how small changes in any of the variables result in changes in the output.

• The input variables \( (x, y, z) \) can represent a physical location or different parameters affecting a system.
• The function's value can represent some quantity of interest, like temperature, pressure, or potential energy.
• Understanding how these functions behave is key to applications in physics, engineering, and economics.

This concept is crucial for dealing with complex systems where an output depends on multiple conditions or variables.

Continuity at a point in a multivariable function means the function behaves smoothly without sudden jumps or breaks at that specific point. For a function \( f \) to be continuous at the point \( (a, b, c) \), two conditions must be true:

• The function's value \( f(a, b, c) \) exists.
• The limit of \( f(x, y, z) \) as \( (x, y, z) \to (a, b, c) \) must be equal to \( f(a, b, c) \).

This ensures that as you approach the point \( (a, b, c) \) from any direction, the function doesn't suddenly change its value. In the context of the given exercise, to show that \( f(x, y, z) = x^2 + y^2 + z^2 \) is continuous at \( (0,0,0) \), we found a suitable \( \delta \) such that the conditions of the epsilon-delta definition are satisfied.
The value of \( \delta \approx 0.1225 \) shows that within this small region around \( (0,0,0) \), the variation in the function's value is limited by \( \epsilon = 0.015 \), confirming the function's smooth behavior at that point.