Definition:Continuous Real Function/Left-Continuous
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Definition
Let $A \subseteq \R$ be an open subset of the real numbers $\R$.
Let $f: A \to \R$ be a real function.
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.
Furthermore, $f$ is said to be left-continuous if and only if:
- $\forall x_0 \in A$, $f$ is left-continuous at $x_0$
Also known as
A function which is left-continuous (either at a point or generally) is also seen referred to as continuous from the left.
Also see
Sources
- 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): $\S 8.6$