kantelfunctione

Kantelfunktione is a term found in some German-language mathematical discussions to describe a class of piecewise linear functions characterized by hinge points, or kinks, in their graphs. The name suggests a focus on the edges or angles where the slope changes.

Formally, a kantelfunktione f: R -> R is continuous and piecewise linear, with a finite set of breakpoints {x1, x2, ..., xk} where the slope changes. Between consecutive breakpoints, the function is affine (linear with a constant slope). The graph typically consists of straight line segments joined at the breakpoints in a way that enforces continuity at those points. Special cases include a single-hinge function, which has exactly one breakpoint, and multi-hinge functions, which exhibit several slope changes.

Key properties include:

- Between breakpoints, f is linear, so derivatives are constant on each interval.

- At breakpoints, the derivative may be discontinuous, producing kinks.

- Convexity of a kantelfunktione depends on the monotonicity of the slopes: nondecreasing slopes yield a convex function.

- Continuity at breakpoints imposes a linear constraint linking the adjacent linear pieces.

Variants and examples:

- A single-hinge function with breakpoint c: f(x) = a1 x + b1 for x < c, and f(x) = a2 x + b2 for x ≥ c with a1 c + b1 = a2 c + b2.

- Multi-hinge constructions can approximate nonlinear shapes by chaining linear pieces.

Applications often appear in statistical modeling (as simple linear surrogates with hinges), numerical approximation, and computational geometry. Related concepts include piecewise linear functions, hinge functions, splines, and ReLU-type activations in machine learning.