Epäjatkuvuusfunktio

Epäjatkuvuusfunktio, also known as a discontinuous function, is a function that is not continuous at one or more points in its domain. A function is considered continuous at a point if the limit of the function as the input approaches that point exists, the function is defined at that point, and the limit equals the function's value at that point. If any of these conditions fail, the function is discontinuous at that point.

There are several types of discontinuities. A removable discontinuity occurs when the limit of the function exists at a point, but the function is either undefined at that point or its value does not equal the limit. This is often represented by a "hole" in the graph. An jump discontinuity happens when the left-hand limit and the right-hand limit at a point exist but are not equal. This results in a sudden "jump" in the function's value. An infinite discontinuity occurs when the limit of the function as it approaches a point is infinite, typically seen in functions with vertical asymptotes.

Examples of discontinuous functions include the Heaviside step function, which jumps from 0 to 1 at x=0, and the Dirichlet function, which is 1 for rational numbers and 0 for irrational numbers, making it discontinuous everywhere. Understanding discontinuities is crucial in various fields of mathematics, including calculus and real analysis, as they can affect the behavior and properties of functions.