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In applied mathematical analysis, "piecewise-regular" 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 (as in a cartoon); [9] a cartoon-like function is a C 2 function, smooth except for the existence of ...
A function property holds piecewise for a function, if the function can be piecewise-defined in a way that the property holds for every subdomain. Examples of functions with such piecewise properties are: Piecewise constant function, also known as a step function; Piecewise linear function; Piecewise continuous function
Since the graph of an affine(*) function is a line, the graph of a piecewise linear function consists of line segments and rays. The x values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots.
The integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f, between a and b.This notion of area fits some functions, mainly piecewise continuous functions, including elementary functions, for example polynomials.
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, Green's function of the Hamiltonian is a key concept with important links to the concept of density of states. The Green's function as used in physics is usually defined with the opposite sign, instead.
The lemma is implicit in the use of piecewise functions. For example, in the book Topology and Groupoids , where the condition given for the statement below is that A ∖ B ⊆ Int A {\displaystyle A\setminus B\subseteq \operatorname {Int} A} and B ∖ A ⊆ Int B . {\displaystyle B\setminus A\subseteq \operatorname {Int} B.}
Define a step function to be a graphon that is piecewise constant, i.e. for some partition of [,], is constant on for all ,. The statement that a graph G {\displaystyle G} has a regularity partition is equivalent to saying that its associated graphon W G {\displaystyle W_{G}} is close to a step function.