Definition:Continuous Real Function/One Side

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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.


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: $\dfrac 1 {1 + e^{1 / x} }$:

\(\displaystyle \lim_{x \mathop \to 0^+} \map f x\) \(=\) \(\displaystyle 0\)
\(\displaystyle \lim_{x \mathop \to 0^-} \map f x\) \(=\) \(\displaystyle 1\)

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.


Also see