Definition:Continuous Real Function/Right-Continuous

Definition

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

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

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.

Furthermore, $f$ is said to be right-continuous if and only if:

$\forall x_0 \in S$, $f$ is right-continuous at $x_0$

A function which is right-continuous (either at a point or generally) is also seen referred to as continuous from the right.

1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 8.6$