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

\(\ds \lim_{x \mathop \to 0^+} \map f x\)

\(=\)

\(\ds 0\)

\(\ds \lim_{x \mathop \to 0^-} \map f x\)

\(=\)

\(\ds 1\)

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.

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity