toefunktsioonide

Toefunktsioonide is a term used in some Estonian-language mathematical literature to denote a class of real-valued functions on a closed interval. A function f: [a,b] → R is called a toefunktsioonide if it is continuous, piecewise linear with a finite number of segments, and the slopes of these segments do not decrease as x increases. In effect, such functions are convex and can be completely described by a finite set of breakpoints and segment slopes.

Properties: Since slopes do not decrease, f is convex, which implies it has a well-defined subgradient at every point and a global minimum that occurs on the set where the derivative crosses from negative to nonnegative. A toefunktsioonide is determined by the positions of breakpoints a = x0 < x1 < ... < xk = b and by the corresponding slopes s1 ≤ s2 ≤ ... ≤ sk+1, with f piecewise linear between breakpoints. The graph is a polygonal chain without curved segments.

Relationship and examples: The class is a subset of convex functions and a subset of piecewise linear functions. Typical examples include f(x) = max{0, x−c} and f(x) = min{d, x} composed to yield piecewise linear convex forms. These functions are widely used in optimization as simple, tractable cost or objective candidates that preserve convexity.

Applications and see also: In operations research and economics, toefunktsioonide can model piecewise linear costs and utilities. In computer graphics, they serve as simple height functions or transfer functions. See also convex function, piecewise linear function, subgradient.