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.
$$
f(x, y)=(x+y) \sin \frac{x}{y}
$$

Key Concepts

Limit in Multivariable Functions

In multivariable calculus, studying limits involves analyzing how a function behaves as its variables approach a particular point. Unlike single-variable limits, the limit in higher dimensions must be the same along every possible path leading to the point. This concept is critical when determining whether a function is continuous at a given point, as the existence of a unique limit is a necessary condition for continuity.

Path Dependence

Path dependence refers to the phenomenon where the value or behavior of a limit of a multivariable function can vary depending on the path taken towards the point of interest. If different paths yield different limits, it indicates that the overall limit does not exist. This analysis is essential in proving discontinuities in multivariable functions and distinguishing between types of discontinuities.

Definition of Continuity

The definition of continuity at a point in multivariable functions stipulates that a function is continuous at a point if the limit of the function as the variables approach the point equals the function's value at that point. This fundamental concept forms the basis for analyzing and classifying discontinuities, ensuring that the function behaves predictably around the point in question.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined at a point or its value at the point does not equal the limit, yet the limit exists and is unique. This type of discontinuity can be 'fixed' by redefining the function at that point so that the function becomes continuous. Recognizing a removable discontinuity involves calculating the limit and determining if the gap in the function can be closed by an appropriate value.

Essential Discontinuity

An essential discontinuity is characterized by a failure of the limit to exist at a point, which may happen because the function takes different limiting values along different paths or behaves erratically near the point. Unlike a removable discontinuity, an essential discontinuity cannot be corrected simply by redefining the function at that point, as the underlying behavior of the function near the point is inherently unstable.

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