Use a graph to determine whether the given function is...

Use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity.
$$
h(x)=\left\{\begin{array}{ll}
\frac{1}{x^{2}} & \text { if } x \neq 0 \\
2 & \text { if } x=0
\end{array}\right.
$$

Key Concepts

Continuity

Continuity is a fundamental concept that determines whether a function behaves without interruption at a point. A function is continuous at a point if the limit of the function as the input approaches that point exists and is equal to the function’s actual value at that point. This concept is critical for understanding the overall smoothness and behavior of a function across its domain.

Limits

Limits describe the value that a function approaches as its input gets arbitrarily close to a specific point. They are essential in the study of continuity because to verify continuity at a point, one must compare the limit as the input approaches that point with the function's value at that same point.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. Such functions require careful analysis, especially at the boundaries where the defining formulas change. The potential for discontinuities is higher at these transition points, making the study of limits and continuity particularly important.

Domain

The domain of a function is the complete set of input values for which the function is defined. When analyzing continuity, it is important to focus on the points within the domain to determine if any points exhibit discontinuous behavior that disrupts the function’s smoothness.

Graphical Analysis

Graphical analysis involves using the visual representation of a function to inspect its behavior. By examining the graph, one can detect discontinuities, such as jumps or breaks, by noting where the graph does not connect smoothly. This method provides an intuitive understanding of the continuity properties of a function.

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