Definition:Continuous Real Function/One Side

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Definition

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 $f \left({x}\right)$ as $x \to x_0$ exists and:

$\displaystyle \lim_{\substack{x \mathop \to x_0^- \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$

where $\displaystyle \lim_{x \mathop \to x_0^-}$ 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 $f \left({x}\right)$ as $x \to x_0$ exists and:

$\displaystyle \lim_{\substack{x \mathop \to x_0^+ \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$

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


Also see