piecewise function

(noun)

A function in which more than one formula is used to define the output over different pieces of the domain.

Related Terms

  • subdomain
  • absolute value

Examples of piecewise function in the following topics:

  • Piecewise Functions

    • Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated intervals.  
    • Example 1: Consider the piecewise definition of the absolute value function:
    •  Substitute those values into the first part of the piecewise function f(x)=x2f(x)=x^2f(x)=x​2​​:
    • Since there is an closed AND open dot at x=1x=1x=1 the function is piecewise continuous there.  
    • When x=2x=2x=2, the function is also piecewise continuous.  
  • Convergence Theorems

    • One has to be a little careful about saying that a particular function is equal to its Fourier series since there exist piecewise continuous functions whose Fourier series diverge everywhere!
    • If fff is piecewise continuous and has left and right derivatives at a point ccc (A right derivative would be: limt→0(f(c+t)−f(c))/t\lim _{t\rightarrow 0} (f(c+t) - f(c)) /tlim​t→0​​(f(c+t)−f(c))/t , t>0t>0t>0 .
    • If fff is continuous with period 2π2\pi2π and f′f'f​′​​ is piecewise continuous, then the Fourier series for fff converges uniformly to fff .
  • Line Integrals

    • A line integral is an integral where the function to be integrated is evaluated along a curve.
    • A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.
    • The function to be integrated may be a scalar field or a vector field.
    • For some scalar field f:U⊆Rn→Rf:U \subseteq R^n \to Rf:U⊆R​n​​→R, the line integral along a piecewise smooth curve C⊂UC \subset UC⊂U is defined as:
    • For a vector field F:U⊆Rn→Rn\mathbf{F} : U \subseteq R^n \to R^nF:U⊆R​n​​→R​n​​, the line integral along a piecewise smooth curve C⊂UC \subset UC⊂U, in the direction of rrr, is defined as:
  • Total Synthesis

    • In order to minimize risk of losing material in a failed reaction and to use time efficiently, a piecewise synthesis scheme is often used.
    • Piecewise synthesis involves breaking target material B into several pieces that can be synthesized separately, and then combining them.
    • For example, a molecule containing an imine, carboxylic acid and ketone will have all three of those functional groups reduced by lithium aluminum hydride.
    • But what if a chemist's intention is to leave the carboxylic acid and ketone functionalities, while reducing the imine?
    • Once the ester is reduced, the acetal can be removed, thus yielding the original ketone functionality.
  • Green's Theorem

    • Let CCC be a positively oriented, piecewise smooth, simple closed curve in a plane, and let DDD be the region bounded by CCC.
    • If LLL and MMM are functions of (x,y)(x,y)(x,y) defined on an open region containing DDD and have continuous partial derivatives there, then:
  • Numerical Integration

    • A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
    • An example of such an integrand is f(x)=exp(x2)f(x) = \exp(x^2)f(x)=exp(x​2​​), the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
    • A 'brute force' kind of numerical integration can be done, if the integrand is reasonably well-behaved (i.e. piecewise continuous and of bounded variation), by evaluating the integrand with very small increments.
    • A definite integral of a function can be represented as the signed area of the region bounded by its graph.
    • Solve for the definite integral of a continuous function over a closed interval
  • Combinations of Capacitors: Series and Parallel

    • To find total capacitance of the circuit, simply break it into segments and solve piecewise .
  • The Divergence Theorem

    • Suppose VVV is a subset of RnR^nR​n​​ (in the case of n=3n=3n=3, VVV represents a volume in 3D space) which is compact and has a piecewise smooth boundary SSS (also indicated with ∂V=S\partial V=S∂V=S).
  • Functional Groups

    • Functional groups are atoms or small groups of atoms (two to four) that exhibit a characteristic reactivity when treated with certain reagents.
    • A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound.
    • Because of their importance in understanding organic chemistry, functional groups have characteristic names that often carry over in the naming of individual compounds incorporating specific groups.
    • In the following table the atoms of each functional group are colored red and the characteristic IUPAC nomenclature suffix that denotes some (but not all) functional groups is also colored.
  • Introduction to Rational Functions

    • A rational function is one such that f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}f(x)=​Q(x)​​P(x)​​, where Q(x)≠0Q(x) \neq 0Q(x)≠0; the domain of a rational function can be calculated.
    • A rational function is any function which can be written as the ratio of two polynomial functions.
    • Any function of one variable, xxx, is called a rational function if, and only if, it can be written in the form:
    • Note that every polynomial function is a rational function with Q(x)=1Q(x) = 1Q(x)=1.
    • A constant function such as f(x)=πf(x) = \pif(x)=π is a rational function since constants are polynomials.
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