In Exercises 18 through 21, prove that the function is...

In Exercises 18 through 21, prove that the function is discontinuous at the origin. Then determine if the discontinuity is removable or essential. If the discontinuity is removable, define $f(0,0)$ so that the discontinuity is removed.
$$
f(x, y)=\frac{x^{2} y^{2}}{x^{2}+y^{2}}
$$

Key Concepts

Continuity in Multivariable Functions

This concept deals with whether a function of more than one variable behaves predictably at a specific point. Specifically, a function is continuous at a point if the limit as the input approaches that point equals the function’s value at that point. In multivariable calculus, verifying continuity requires considering all possible paths toward the point in question.

Limits of Functions of Several Variables

In multivariable calculus, determining if a limit exists requires showing that the limit is the same along every possible path approaching the point. If even one path yields a different limit, the overall limit does not exist. This concept underpins the analysis of continuity and discontinuity for functions of two or more variables.

Path Dependence in Limit Evaluation

This idea involves checking the limit of a function as the inputs approach the specified point along different paths. If various paths – linear, parabolic, or any other curve – yield different limit values, it indicates that the overall limit does not exist. This is a key tool in diagnosing discontinuities in multivariate functions.

Removable versus Essential Discontinuities

A removable discontinuity occurs when the limit of a function exists at a point, but the function is either not defined or not properly defined at that point. It can be 'fixed' by redefining the function at the discontinuity. In contrast, an essential (or non-removable) discontinuity occurs when the limit does not exist or differs depending on the path of approach.

Polar Coordinate Transformation in Limit Problems

Using polar coordinates is a useful technique when evaluating multivariable limits, especially when the function involves expressions like x² + y². This transformation can simplify the limit expression by consolidating the variables into a single radius term, thus providing clearer insight into the behavior of the function as the radius approaches zero.

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