Kfunktio

Kfunktio is a parametric mathematical function used to illustrate bounded, monotone nonlinear mappings on the real numbers. For a nonnegative parameter K, it is defined by f_K(x) = x / (1 + K |x|). The function is continuous and strictly increasing for all x, with the special case K = 0 reducing to the identity f_0(x) = x. When K > 0, the function saturates for large |x|, approaching ±1/K as x → ±∞.

Key properties of Kfunktio include its continuity and differentiability across all real numbers, with a positive derivative everywhere. The derivative is f_K'(x) = 1/(1 + K|x|)^2, which yields a Lipschitz constant of 1. This makes Kfunktio a simple, well-behaved bounded activation-like nonlinearity that preserves order while preventing unbounded growth.

Variants of the concept generalize the form by introducing additional parameters or powers, such as f_K,α(x) = x / (1 + K |x|)^α. These forms retain monotonicity and boundedness, while allowing different saturation rates. In practice, Kfunktio and its variants are discussed in theoretical treatments of Lipschitz mappings and as illustrative bounded nonlinearities in neural networks and robust statistics.

Applications for Kfunktio include serving as a lightweight activation function in neural networks, where bounded outputs help stabilize training, and as an example of a monotone, saturating nonlinearity in mathematical analyses. Related concepts include activation functions in general, as well as Lipschitz continuity and bounded nonlinear mappings.