Definition:Piecewise Continuous Function/One-Sided Limits

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 $\ds \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\ds \lim_{x \mathop \to {x_i}^-} \map f x$ exist.

Also known as

Some sources use sectionally continuous for piecewise continuous function.

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

• Piecewise Continuous Function with One-Sided Limits is Darboux Integrable

• Piecewise Continuous Function with One-Sided Limits is Uniformly Continuous on Each Piece

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.): $\S 6.2$