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

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Key Concepts

Removable Versus Essential Discontinuity

A removable discontinuity is present when the limit exists at a point but the function is either undefined or assigned a different value; it can be 'fixed' by appropriately redefining the function at that point. In contrast, an essential discontinuity occurs when the limit does not exist or is inherently inconsistent due to path-dependence or other factors.

Discontinuity

A discontinuity occurs at a point where a function is not continuous. In multivariable contexts, this can happen due to issues such as the limit not existing uniformly across all paths or the function being undefined at that point.

Continuity in Multivariable Functions

Continuity at a point for functions of several variables requires that the limit as the variables approach that point equals the function's value at that point. Any deviation, such as an undefined point or different limits along different paths, indicates a lack of continuity.

Multivariable Limits

In functions of several variables, the limit as (x, y) approaches a point exists only if the function values approach the same number regardless of the path taken. This concept is crucial when verifying whether a limit exists at a given point, especially when different paths yield different results.

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