In Exercises 8 through 17, determine the region of...

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{x y}{\sqrt{16-x^{2}-y^{2}}}

Key Concepts

Inequalities and Boundary Sets in ?²

Determining the region of continuity in the plane often involves solving inequalities that describe geometric regions, such as circles or other curves. These inequalities delineate the boundaries within which the function is defined and continuous, and they are essential when producing a sketch of the region.

Square Root Function Domains

When a function involves an even-indexed square root, the expression inside the square root (the radicand) must be non-negative to yield real number outputs. This creates a constraint on the input values, forming part of the domain.

Restrictions from Denominators

In any function involving division, it is necessary to ensure that the denominator is not zero. Locations where the denominator becomes zero must be excluded from the domain to maintain the function's definition and continuity.

Domain of a Function

In the context of functions of several variables, the domain represents all combinations of input values for which the function is well-defined. This involves identifying any restrictions, such as avoiding negative values under an even-indexed root or division by zero conditions.

Continuity in Multivariable Functions

A multivariable function is continuous at a point if it is defined at that point and the limit as the variables approach that point equals the function's value there. The region of continuity is the set of all such points in the domain, which must be determined by considering the properties of the individual operations within the function.

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