One-Sided Continuity/Examples

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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.