Grenzebenen

Grenzebenen, also known as boundary planes or limit planes, are a concept in mathematics, particularly in the field of geometry and topology. They are used to describe the behavior of functions and sets at their boundaries or limits. In a more technical sense, a Grenzebene can be defined as a plane that is tangent to a surface at a point where the surface has a sharp edge or corner. This concept is often used in the study of manifolds and differential geometry.

Grenzebenen are important in the study of continuity and differentiability of functions. A function is said to be continuous at a point if its value at that point is equal to its limit as the input approaches that point. Similarly, a function is differentiable at a point if its derivative at that point is equal to the limit of the difference quotient as the input approaches that point. In both cases, Grenzebenen can be used to describe the behavior of the function at its boundaries or limits.

Grenzebenen are also used in the study of convex sets and convex functions. A convex set is a set that contains the line segment between any two of its points. A convex function is a function whose epigraph, the set of points on or above its graph, is a convex set. Grenzebenen can be used to describe the behavior of convex functions at their boundaries or limits.

In summary, Grenzebenen are a fundamental concept in mathematics, used to describe the behavior of functions and sets at their boundaries or limits. They are important in the study of continuity, differentiability, convex sets, and convex functions.