- Piecewise
In
mathematics , a piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of theindependent variable . Mathematically, a real-valued function "f" of a realvariable "x" is a relationship whose definition is given differently on disjoint subsets of its domain (known as subdomains).The word "piecewise" is also used to describe any property of a piecewise-defined function that holds for each piece but may not hold for the whole
domain of the function. A function is piecewise differentiable or piecewise continuously differentiable if each piece is differentiable throughout its domain. 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.Notation and interpretation
Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. For example, consider the piecewise definition of the
absolute value function::For all values of "x" less than zero, the first function (−"x") is used, which negates the sign of the input value, making negative numbers positive. 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.Consider the piecewise function "f"("x") evaluated at certain values of "x":
Thus, in order to evaluate a piecewise function at a given input value, the appropriate subdomain needs to be chosen in order to select the correct function and produce the correct output value.
Continuity
A piecewise function is continuous on a given interval if it is defined throughout that interval, its appropriate constituent functions are continuous on that interval, and there is no discontinuity at each endpoint of the subdomains within that interval.
The pictured function, for example, is not continuous throughout its domain, though its constituent functions are continuous, because there exists a discontinuity at "x" = "x"0, which is likely the common endpoint of two subdomains.
Common examples
*
Absolute value
*Heaviside step function
*Piecewise linear function
*Spline
*B-spline
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