Definition:Continuous Real Function/One Side
Definition
Let $A \subseteq \R$ be an open subset of the real numbers $\R$.
Let $f: A \to \R$ be a real function.
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 $\map f x$ 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.
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 $\map f x$ 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.
Examples of One-Sided Continuity
Example: $\dfrac 1 {1 + e^{1 / x} }$ at $x = 0$
Consider the real function $f$ defined as:
- $f := \dfrac 1 {1 + e^{1 / x} }$
Then from One-Sided Limit of Real Function: Examples: $\dfrac 1 {1 + e^{1 / x} }$:
| \(\ds \lim_{x \mathop \to 0^+} \map f x\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \lim_{x \mathop \to 0^-} \map f x\) | \(=\) | \(\ds 1\) |
Hence however $\map f 0$ is defined, $f$ cannot be made to be continuous at $x = 0$.
However, let us define $g$ as:
- $g := \begin {cases} \dfrac 1 {1 + e^{1 / x} } & : x \ne 0 \\ 0 & : x = 0 \end {cases}$
Then $g$ is right-continuous.
Similarly, let us define $h$ as:
- $h := \begin {cases} \dfrac 1 {1 + e^{1 / x} } & : x \ne 0 \\ 1 & : x = 0 \end {cases}$
Then $h$ is left-continuous.