piecewisefunctie

Piecewisefunctie, or piecewise function in English, is a function that is defined by different expressions on different parts of its domain. Typically the domain is a subset of the real numbers, and it is partitioned into subintervals. On each interval Di a specific formula fi(x) is used, so f(x) = fi(x) for x in Di. The value at boundary points is determined by the piece that includes that point, though some definitions may assign the boundary to a different piece.

A common way to express a piecewisefunctie is with a compact notation, for example f(x) = { x^2 for x < 0; 2x for x ≥ 0 }. This means f equals x^2 when x is negative and equals 2x when x is nonnegative. Such definitions enable simple modeling of regimes that change abruptly at certain points, such as thresholds or switches between rules.

Properties and behavior at boundaries are important. The graph consists of the graphs of the individual pieces, joined at the boundary points. If the left and right limits agree and match the function's value, the function is continuous at that boundary. Differentiability at the boundary may fail if the left and right derivatives do not agree, producing a corner or cusp.

Applications are widespread. Piecewisefuncties appear in modeling piecewise phenomena, absolute value expressions, tax brackets, and various forms of piecewise-linear approximations. They are also common in programming and mathematics education as a straightforward way to describe rules that change with the input.