Clipfunktiota

Clipfunktiota, also known as the clipping function or clamp function, is a mathematical function that restricts its input to a specified closed interval [a, b]. Formally, clip_{a,b}(x) = min(max(x, a), b). Equivalently: a if x < a; x if a <= x <= b; b if x > b.

Properties of the function include that it is monotone nondecreasing and 1-Lipschitz, meaning that it does not expand distances: |clip(x) - clip(y)| <= |x - y|. It is piecewise linear with slopes 0 or 1. It is differentiable almost everywhere; the derivative is 0 for x < a or x > b, and 1 for a < x < b, with non-differentiability at x = a and x = b. On inputs within the interval, clip(x) = x (the function is idempotent on [a, b]).

Generalizations and variants include vector clipping, where each component is clipped to its own interval: clip(x)_i = clip_{a_i,b_i}(x_i). The concept also extends to higher dimensions and to different bound schemes per coordinate or norm-based constraints.

Applications are widespread in engineering and computing. They are used to prevent saturation in signal processing and data quantization, clamp pixel values in image processing, and limit actuator commands in control systems. In neural networks, clipped activations such as hard tanh employ a similar idea as a simple nonlinearity. In programming, clip is often implemented via min and max utilities to enforce bounds.

Examples: clip_{0,1}(0.5) = 0.5; clip_{0,1}(-0.2) = 0; clip_{0,1}(1.5) = 1.