# Definition:Piecewise Continuous Function/One-Sided Limits

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## Contents

## Definition

Let $f$ be a real function defined on a closed interval $\closedint a b$.

$f$ 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**.

Some sources refer to condition $(2)$ as that $\map f x$ is **finite at the endpoints**, but $\mathsf{Pr} \infty \mathsf{fWiki}$ demands more rigor in its use of the term **finite**.

The one-sided limits can also be seen denoted as:

- $\map f {x_{i − 1} + 0}$ and $\map f {x_i - 0}$

## Also see

- Piecewise Continuous Function with One-Sided Limits is Bounded
- Bounded Piecewise Continuous Function may not have One-Sided Limits

## Sources

- 1961: I.N. Sneddon:
*Fourier Series*... (previous) ... (next): Chapter Two: $\S 1$. Piecewise-Continuous Functions - 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Sectional or Piecewise Continuity - 2007: Tyn Myint-U and Lokenath Debnath:
*Linear Partial Differential Equations for Scientists and Engineers*(4th ed.): $\S 6.2$