Crucially, in most settings, there must only be a 'finite' number of subdomains, each of which must be an interval, in order for the overall function to be called ' piecewise'. Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function which is defined by multiple sub-functions. A piecewise function is a function built from pieces of different. For example, 'If x<0, return 2x, and if x0, return 3x.' These are called piecewise functions, because their rules arent uniform, but consist of multiple pieces. In applied mathematical analysis, piecewise functions have been found to be consistent with many models of the human visual system, where images are perceived at a first stage as consisting of smooth regions separated by edges. Some functions have simple rules, like 'for every x, return xВІ.' However, there can be other rules that are more elaborate. For example, a piecewise polynomial function is a function that is a polynomial on each of its sub-domains, but possibly a different one on each. Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function.

In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain (a sub-domain). Specific instances of piecewise functions include: For all values of 'x' greater than or equal to zero, the second function ('x') is used, which evaluates trivially to the input value itself. The pictured function contains a jump discontinuity at formula_2.ДЇor all values of 'x' less than zero, the first function (в€’'x') is used, which negates the sign of the input value, making negative numbers positive. The pictured function, for example, is piecewise continuous throughout its subdomains, but is not continuous on the entire domain.

In particular, shearlets have been used as a representation system to provide sparse approximations of this model class in 2D and 3D. Although the 'pieces' in a piecewise definition need not be intervals, a function is not called ' piecewise linear' or ' piecewise continuous' or ' piecewise differentiable' unless the pieces are intervals.ДЄ piecewise function is continuous on a given interval if the following conditions are met: In convex analysis, the notion of a derivative may be replaced by that of the subderivative for piecewise functions. A function is piecewise differentiable or piecewise continuously differentiable if each piece is differentiable throughout its subdomain, even though the whole function may not be differentiable at the points between the pieces. The word ' piecewise' is also used to describe any property of a piecewise-defined function that holds for each piece but not necessarily hold for the whole domain of the function.

Definition piecewise function