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

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 $f(x)=x^2$:
    • Since there is an closed AND open dot at $x=1$ the function is piecewise continuous there.  
    • When $x=2$, 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 $f$ is piecewise continuous and has left and right derivatives at a point $c$ (A right derivative would be: $\lim _{t\rightarrow 0} (f(c+t) - f(c)) /t$ , $t>0$ .
    • If $f$ is continuous with period $2\pi$ and $f'$ is piecewise continuous, then the Fourier series for $f$ converges uniformly to $f$ .

  • 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 $f:U \subseteq R^n \to R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as:
    • For a vector field $\mathbf{F} : U \subseteq R^n \to R^n$, the line integral along a piecewise smooth curve $C \subset U$, in the direction of $r$, 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 $C$ be a positively oriented, piecewise smooth, simple closed curve in a plane, and let $D$ be the region bounded by $C$.
    • If $L$ and $M$ are functions of $(x,y)$ defined on an open region containing $D$ 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 $f(x) = \exp(x^2)$, 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 $V$ is a subset of $R^n$ (in the case of $n=3$, $V$ represents a volume in 3D space) which is compact and has a piecewise smooth boundary $S$ (also indicated with $\partial V=S$).

  • 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 $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; 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, $x$, 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 $Q(x) = 1$.
    • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.