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.
$$
h(x)=\left\{\begin{array}{ll}
\frac{|x|}{x} & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
$$

Key Concepts

Graphical Analysis

Graphical analysis involves plotting the function and visually inspecting the graph to determine whether it is seamless across its domain. This method is useful for identifying discontinuities such as jumps or holes, which appear as breaks in the graph, and confirming the smooth transition of function values.

Limits

The concept of limits is fundamental to the analysis of continuity. Limits examine the behavior of a function as the input approaches a certain value from either side. For a function to be continuous at a point, the limit from the left and the right must exist and be equal to the function's value at that point.

Discontinuities

Discontinuities are points at which a function fails to be continuous. They can arise when the left-hand and right-hand limits at a point differ (jump discontinuity), when the limit exists but is not equal to the function's value (removable discontinuity), or when the limit does not exist at all. Identifying these points is essential for understanding the function's overall behavior.

Continuity

Continuity is the property of a function where small changes in the input produce small changes in the output, without any abrupt jumps, breaks, or holes in the graph. Formally, a function is continuous at a point if the limit as the input approaches that point equals the function’s value at that point, and it is continuous on a domain if this condition holds for every point in that domain.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domains. This means that verifying their continuity requires checking not only the continuity within each segment but also the behavior at the boundaries where the pieces meet, ensuring that the left-hand and right-hand limits at the joining points are equal to each other and, where applicable, to the function's value.

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