Definition:Continuous Real Function/Left-Continuous

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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