# Definition:Continuous Real Function/Left-Continuous/Point

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## Contents

## 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:

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

where $\displaystyle \lim_{x \mathop \to x_0^-}$ 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

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.2$: Examples - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 8.6$