Definition:Piecewise Continuous Function

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Definition

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.


$f$ is piecewise continuous if and only if:

there exists a finite subdivision $\left\{ {x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that:
for all $i \in \left\{ {1, 2, \ldots, n}\right\}$, $f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$.


Further conditions can be placed on this definition to add specialisation as necessary:


With Improper Integrals

$f$ is piecewise continuous with improper integrals if and only if:

there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that for all $i \in \left\{ {1, 2, \ldots, n}\right\}$:
$(1): \quad f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$
$(2): \quad$ the improper integrals $\displaystyle \int_{ {x_{i - 1} }^+}^{ {x_i}^-} f \left({x}\right) \rd x$ all exist.


Bounded

$f$ is a bounded piecewise continuous function if and only if:

there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that:
$(1): \quad$ for all $i \in \left\{{1, 2, \ldots, n}\right\}$, $f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$
$(2): \quad$ $f$ is bounded on $\left[{a \,.\,.\, b}\right]$.


With One-Sided Limits

$f$ is piecewise continuous with one-sided limits if and only if:

there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that, for all $i \in \set {1, 2, \ldots, n}$:
$(1): \quad f$ is continuous on $\openint {x_{i − 1} } {x_i}$
$(2): \quad$ the one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map f x$ exist.


Also known as

Some sources hyphenate: piecewise-continuous.


Also defined as


There are other definitions of Piecewise Continuous Function. For example, the following variations exist:

  • $f$ need not be defined at the points $x_i$.
  • The subdivision above can be infinite when the domain of $f$ is unbounded.
  • The codomain of $f$ is $\C$ instead of $\R$.


Also see



Comments

Possible properties of piecewise continuous functions:

  • It seems obvious that a linear combination, a product, a quotient, and a composite of piecewise continuous functions are piecewise continuous functions.
  • Also, a function piecewise continuous on adjacent intervals should be piecewise continuous on the union of these intervals.


Sources