An asymptotic discontinuity is a type of mathematical discontinuity that occurs when a function approaches a specific value or point, but does not actually reach that value. In other words, the function approaches a particular limit, but there is a gap or jump in its behavior at that point.
Asymptotic discontinuities typically arise when there is a vertical asymptote or a removable discontinuity in a function. A vertical asymptote occurs when the function approaches infinity or negative infinity as the input approaches a particular value. At this point, the function may have a gap or jump in its graph.
For example, consider the function f(x) = 1/x. As x approaches 0 from the positive side, the function approaches positive infinity, and as x approaches 0 from the negative side, the function approaches negative infinity. This creates a vertical asymptote at x = 0, and the function has an asymptotic discontinuity at that point.
It's important to note that asymptotic discontinuities are different from other types of discontinuities such as jump discontinuities or removable discontinuities, where the function has a finite gap or a removable point at a specific value. Asymptotic discontinuities refer specifically to the behavior of a function as it approaches a limit without actually reaching it.