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A jump discontinuity occurs when there is a sudden change in the value of a function at a certain point. Let's consider the function at the point \( x = 22 \), which has a jump discontinuity. This means that the left-hand limit and right-hand limit exist, but they are not equal. In other words, as you approach \( x = 22 \) from the left, the function may approach a certain value, while approaching from the right, it may approach a completely different value. Imagine standing at the edge of a cliff, looking down to a lower ledge just to the side; that's essentially how a jump discontinuity behaves on the graph.

Here are some characteristics of a jump discontinuity to help visualize it better:

• The function values abruptly "jump" to a different height at the discontinuous point.
• On a graph, you will see separate dots or gaps at the discontinuity.
• Both left-hand and right-hand limits exist but are not the same.

For example, a common visual for this is having the left-hand portion end abruptly and the right-hand portion resume at a different level.

A removable discontinuity is where a function is not continuous at a specific point, yet it is possible to "fill in the gap" to make it continuous. It typically occurs when the function has a hole at a particular \( x \)-value. This concept wasn't needed directly in the original exercise, but is good to understand in relation to discontinuities.

Here’s how you can spot a removable discontinuity:

• A hole in the function appears at an isolated point.
• The function can be redefined at the missing point to make it continuous.
• On a graph, it appears as a missing point that can often match the expected limit from either side.

Typically, if you find that the limit of the function as \( x \) approaches a certain value from both sides is the same, but the function isn't defined at that exact \( x \)-value, it's a removable discontinuity. A simple shift in the definition at the point could make the entire function smooth.

The left-hand limit is a crucial concept when examining the behavior of functions as we approach a specific point from the left side. Specifically, when we talk about a function being continuous from one side, the left-hand limit comes into play. This was significant in the original exercise at \( x = 2 \), where the function is continuous only from the left.

Here's what you need to know about the left-hand limit:

• The notation is \( \lim_{{x \to a^-}} f(x) \), indicating the limit as \( x \) approaches \( a \) from the left.
• If the left-hand limit equals the function's value at \( a \), the function is continuous from the left at \( a \).
• For the function in the exercise, at \( x = 2 \), this means that from values less than 2, the path of the function matches at the point.

In practical terms, imagine walking towards a door; the left-hand limit defines how close you can get to the door from the left without crossing it.

The right-hand limit complements the left-hand limit by describing the behavior of a function as it approaches a specific point from the right. It helps identify the continuity and value a function achieves as it nears a particular \( x \)-value. Although the original exercise focused on left-hand continuity at \( x = 2 \) and did not emphasize the right-hand limit, understanding it is important for comprehensive knowledge.

Some key points about the right-hand limit include:

• The notation is \( \lim_{{x \to a^+}} f(x) \), showing the limit as \( x \) draws near \( a \) from the right.
• If the right-hand limit equals the function's value at \( a \), the function is right continuous at \( a \).
• In cases like \( x = 22 \) from the exercise, where the right-hand limit doesn't match the left-hand limit, we encounter a jump discontinuity.

Visualize coming towards a point on a graph from the right-moving direction: the right-hand limit specifies the value you are approaching as you get close to that point.