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In calculus, a jump discontinuity occurs when there is an abrupt change in the value of a function at a specific point. This happens because the left-hand limit and the right-hand limit of the function at that point are not equal. Imagine walking on a path and suddenly encountering a vertical step up or down. That's similar to a jump discontinuity in a graph.

If we have a function that is defined differently on either side of a point, it can cause this type of discontinuity. For instance, using the original problem, consider the function:

• For \( x < 2 \), the function is defined as \( f(x) = x + 1 \).
• For \( x > 2 \), it is \( f(x) = x - 1 \).

At \( x = 2 \), the function jumps from 3 to 1, resulting in a vertical gap at that point. Each side of the point follows a different rule, leading to the discontinuity.

A removable discontinuity is like finding a small hole on a path that can be easily patched up. It's a point on the graph where the function isn't continuous, but there's an easy fix to make it continuous. This can occur when the limit of the function as it approaches a specific point exists, but it doesn’t align with the function’s value at that point.

To visualize this, consider the example of the function from the original exercise:

• As \( x \) approaches 4 from either direction, the function \( f(x) = x \). This means the limit as \( x \to 4 \) is 4.
• However, the value of the function specifically defined at \( x = 4 \) is \( f(4) = 0 \).

The function thus has a removable discontinuity at \( x = 4 \) because we can "remove" this discrepancy by redefining \( f(4) \) to be 4, aligning it with the limit.

Graph sketching involves plotting the behavior of a function on a coordinate plane. This helps to visualize concepts like discontinuities. To graph the function with discontinuities as described in the original exercise:

• Begin by sketching the line \( y = x + 1 \) for \( x < 2 \), keeping it slightly above the line \( y = x \). Place a distinct point at (2, 3) to mark the end of this segment.
• Next, draw the line \( y = x - 1 \) for \( x > 2 \), beginning just after \( x = 2 \).
• Continue this line until just before \( x = 4 \), where you make a small open circle (hole) at \((4, 4)\) to indicate the removable discontinuity.
• Finally, plot the point (4, 0) to show the actual defined value at \( x = 4 \).

This method of graph sketching illustrates both the jump and removable discontinuities effectively, allowing for a clear understanding of their characteristics.