# One-Sided Continuity/Examples

## Examples of One-Sided Continuity

### Example: $\dfrac 1 {1 + e^{1 / x} }$ at $x = 0$

Consider the real function $f$ defined as:

$f := \dfrac 1 {1 + e^{1 / x} }$
 $\ds \lim_{x \mathop \to 0^+} \map f x$ $=$ $\ds 0$ $\ds \lim_{x \mathop \to 0^-} \map f x$ $=$ $\ds 1$

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.