Show that the function defined by f(x, y, z)=(y+1) (x^2-z^2)/(x^2+z^2) for (x, y, z) ≠(0,0,0)

step 1 Here, we're given the function of f, comma, x, y, and z to be equal to y plus 1 times x squared

Show that the function defined by f(x, y, z)=(y+1)...

Key Concepts

Piecewise Defined Functions and Continuity

When a function is defined by different expressions in different regions of its domain, special attention must be given to the boundary or the point where the definitions change. In order for the function to be continuous at that point, the limit as the variables approach the point must equal the value defined at the point. A discrepancy between these values indicates discontinuity.

Coordinate Transformation in Limit Analysis

Using coordinate transformations, such as polar or spherical coordinates, can simplify the limit analysis of multivariable functions. These transformations isolate the radial and angular components, making it easier to determine whether the limit depends on the direction of approach, which is crucial in establishing whether a limit—and hence continuity—exists.

Continuity in Multivariable Functions

In the context of functions of several variables, a function is continuous at a point if the limit of the function as the input approaches that point from all possible directions exists and equals the function’s value at that point. This concept extends one-dimensional continuity to higher dimensions, where the behavior of the function may vary depending on the path taken to approach the point.

Path-Dependent Limit

A key method in analyzing continuity in multivariable functions is to examine the limit along different paths. If the limit of the function varies when approached along different trajectories, then the overall limit does not exist. This is a common approach to prove that a function is not continuous at a given point.

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