In Exercises 23 through 26, use the definitions and theorems...

In Exercises 23 through 26, use the definitions and theorems of Exercise 22 to discuss the continuity of the given function.
$$
f(x, y, z)= \begin{cases}\frac{x z-y^{2}}{x^{2}+y^{2}+z^{2}} & \text { if }(x, y, z) \neq(0,0,0) \\ 0 & \text { if }(x, y, z)=(0,0,0)\end{cases}
$$

Key Concepts

Continuity in Multiple Variables

In multiple dimensions, a function is continuous at a point if the limit exists when approaching that point from any direction and is equal to the function's value at that point. This entails using the epsilon?delta definition or sequential criteria to rigorously confirm that arbitrarily small changes in the input result in small changes in the output.

Definition of Limit in R?

The limit of a multivariable function at a point is defined by requiring that for every sequence converging to that point, the corresponding sequence of function values converges to the same limit. Equivalently, using the epsilon-delta approach, for every epsilon greater than zero, there exists a delta such that the distance between (x, y, z) and the point is less than delta implies that the absolute difference between the function value and the limit is less than epsilon.

Piecewise Function Analysis

Analyzing functions that are defined piecewise necessitates special attention at the point where the definition changes. It is important to verify that the limit as the variable approaches that point from all directions is equal to the value assigned by the function at the point, thereby ensuring continuity throughout.

Path Dependence in Multivariable Limits

Because multivariable limits must hold true regardless of the path taken towards a point, checking limits along various paths is a common strategy. If the limit varies along different paths, the overall limit does not exist, indicating a discontinuity or a potential removable discontinuity.

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