Stetigkeitsarten

Stetigkeitsarten refers to the different ways a function can be continuous or discontinuous. Continuity is a fundamental concept in calculus and analysis, describing whether a function's graph can be drawn without lifting the pen.

A function is considered continuous at a point if three conditions are met: the function is defined at that point, the limit of the function exists 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. Removable discontinuities occur when a hole exists in the graph, and the function can be made continuous by redefining the function's value at that single point. These are often caused by a factor that cancels out in the numerator and denominator of a rational function.

Jump discontinuities happen when the left-hand limit and the right-hand limit at a point exist but are not equal. This creates a "jump" in the graph. Infinite discontinuities arise when the limit of the function approaches infinity or negative infinity at a point, typically associated with vertical asymptotes.

Essential or non-removable discontinuities are those that do not fall into the categories of removable or jump discontinuities, and often involve more complex behavior such as oscillations. Understanding these Stetigkeitsarten is crucial for analyzing function behavior, solving equations, and developing mathematical models.