Category:Examples of One-Sided Continuity

This category contains examples of One-Sided Continuity.

Let $A \subseteq \R$ be an open subset of the real numbers $\R$.

Let $f: A \to \R$ be a real function.

Continuity from the Left at a Point

Let $x_0 \in A$.

Then $f$ is said to be left-continuous at $x_0$ if and only if the limit from the left of $\map f x$ as $x \to x_0$ exists and:

$\ds \lim_{\substack {x \mathop \to x_0^- \\ x_0 \mathop \in A} } \map f x = \map f {x_0}$

where $\ds \lim_{x \mathop \to x_0^-}$ is a limit from the left.

Continuity from the Right at a Point

Let $x_0 \in S$.

Then $f$ is said to be right-continuous at $x_0$ if and only if the limit from the right of $\map f x$ as $x \to x_0$ exists and:

$\ds \lim_{\substack {x \mathop \to x_0^+ \\ x_0 \mathop \in A}} \map f x = \map f {x_0}$

where $\ds \lim_{x \mathop \to x_0^+}$ is a limit from the right.