Understanding Jump Discontinuity in Mathematics | Causes, Examples, and Mathematical Expressions

f has a jump discontinuity at x=a

A jump discontinuity occurs when a function has a sudden, non-removable jump in its graph at a specific x-value, denoted by “a” in this case

A jump discontinuity occurs when a function has a sudden, non-removable jump in its graph at a specific x-value, denoted by “a” in this case. At this point, the function does not have a defined value or limit from either the left or right side of the x-value.

To provide a more detailed explanation, let’s consider an example. Suppose we have a function f(x) defined as follows:

f(x) =
-2 for x < a 3 for x > a

In this case, when x approaches the x-value ‘a’ from the left side (i.e., as x gets closer to ‘a’ from smaller values of x), the function takes the value -2. However, when x approaches ‘a’ from the right side (i.e., as x gets closer to ‘a’ from larger values of x), the function takes the value 3. Thus, at x=a, the function has a sudden jump from -2 to 3, creating a jump discontinuity.

Mathematically, we can express this jump discontinuity using limits. The left-hand limit (LHL) of f(x) as x approaches ‘a’ is -2, and the right-hand limit (RHL) of f(x) as x approaches ‘a’ is 3. However, since the LHL and RHL do not agree (i.e., -2 ≠ 3), the function does not have a regular limit at x=a, resulting in the jump discontinuity.

It’s important to note that jump discontinuities can occur in various forms and shapes, depending on the specific function. The example provided is just one possibility, but jump discontinuities can also have vertical asymptotes, holes, or other characteristics.

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