Final answer:
Jump discontinuity in a mathematical function is when there is a sudden change in the function's value at a certain point, and the limits approaching from the left and right are not equal. It is a nonremovable discontinuity and is common in piecewise functions such as step functions.
Step-by-step explanation:
A jump discontinuity occurs in a function when there is a sudden change in the function's value at a point in its domain. This type of discontinuity is considered nonremovable because the limits from the left and the right of the discontinuity are not equal. More formally, a function f(x) has a jump discontinuity at a point x = c if the following conditions are met:
- The limits from the left (lim x←c- f(x)) and from the right (lim x→c+ f(x)) exist but are not equal.
- The function is defined at x = c either directly, or it has distinct left-hand and right-hand values at c.
Jump discontinuities are common in piecewise functions where different rules apply to different intervals of the function's domain. An example is the step function, which jumps from one value to another at certain points. The point where the function jumps is where the discontinuity occurs.
To understand the concept of jump discontinuity, consider a simple piecewise function defined as:
- f(x) = 1 for x < 2
- f(x) = 2 for x ≥ 2
At x = 2, the function value suddenly changes from 1 to 2. Hence, it exhibits a jump discontinuity at that point.