One-Sided Continuity/Examples/Reciprocal of 1 + e to the Reciprocal of x

Example of One-Sided Continuity
Consider the real function $f$ defined as:
 * $f := \dfrac 1 {1 + e^{1 / x} }$

Then from One-Sided Limit of Real Function: Examples: $\dfrac 1 {1 + e^{1 / x} }$:

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:


 * $h := \begin {cases} \dfrac 1 {1 + e^{1 / x} } & : x \ne 0 \\ 0 & : x = 1 \end {cases}$

Then $h$ is left-continuous.