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 $f$.
$$
f(x, y)= \begin{cases}\frac{\sin (x+y)}{x+y} & \text { if } x+y \neq 0 \\ 1 & \text { if } x+y=0\end{cases}
$$

Key Concepts

Piecewise Defined Functions

A piecewise defined function is one that is described by different expressions depending on the region of the domain being considered. In the context of functions of multiple variables, each piece must be examined for continuity at the boundaries where the definitions switch. The behavior at the junctions or boundaries is crucial in determining if the function can be smoothly defined throughout its domain, or if any discontinuities exist due to the switching between definitions.

Removable Discontinuity

Removable discontinuities occur when a function is not initially defined at a point, or its definition leads to a break in continuity, even though the limit exists at that point. The discontinuity can be 'removed' by redefining the function at that point to equal the limit. This concept is critical when assessing the continuity of a piecewise function where one of the pieces might have an indeterminate form that can be resolved by proper definition at a specific boundary, ensuring the function remains continuous across that boundary.

Continuity of Multivariable Functions

This concept involves determining whether a function defined on ?² (or higher dimensions) is continuous at a given point by checking if the limit of the function as it approaches that point equals the function's value at that point. In multivariable calculus, continuity requires that the function behaves predictably along every possible path of approach, not just along the coordinate axes. This is essential when analyzing the overall region where the function maintains a consistent behavior without any abrupt changes or breaks.

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