Bounded Piecewise Continuous Function may not have One-Sided Limits

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Theorem

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$, $a < b$.


Let $f$ be a bounded piecewise continuous function.

$f$ is a bounded piecewise continuous function if and only if:

there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that:
$(1): \quad$ for all $i \in \left\{{1, 2, \ldots, n}\right\}$, $f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$
$(2): \quad$ $f$ is bounded on $\left[{a \,.\,.\, b}\right]$.


Then it is not necessarily the case that $f$ is a piecewise continuous function with one-sided limits:

$f$ is piecewise continuous with one-sided limits if and only if:

there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that, for all $i \in \set {1, 2, \ldots, n}$:
$(1): \quad f$ is continuous on $\openint {x_{i − 1} } {x_i}$
$(2): \quad$ the one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map f x$ exist.


Proof

Consider the function:

$f \left({x}\right) = \begin{cases} 0 & : x = a \\ \sin \left({\dfrac 1 {x - a} }\right) & : x \in \left({a \,.\,.\, b}\right] \end{cases}$

Consider the (finite) subdivision $\left\{{a, b}\right\}$ of $\left[{a \,.\,.\, b}\right]$.

We observe that $\sin \left({\dfrac 1 {x - a} }\right)$ is continuous on $\left({a \,.\,.\, b}\right)$.

Since $f \left({x}\right) = \sin \left({\dfrac 1 {x - a} }\right)$ on $\left({a \,.\,.\, b}\right)$, it follows that $f$ is continuous on $\left({a \,.\,.\, b}\right)$.

Also, $f$ is bounded by the bound $1$ on $\left[{a \,.\,.\, b}\right]$.

Therefore $f$ is a bounded piecewise continuous function on the closed interval $\left[{a \,.\,.\, b}\right]$.


We now investigate whether $f$ has one-sided limits at the endpoints of $\left[{a \,.\,.\, b}\right]$.

The function $\sin \left({\dfrac 1 {x - a} }\right)$ varies between $-1$ and $+1$ as $x$ approaches $a$ from above.

Thus it does not converge.

Since $f \left({x}\right) = \sin \left({\dfrac 1 {x - a} }\right)$ when $x > a$ we conclude that $\displaystyle \lim_{x \mathop \to a^+} f \left({x}\right)$ does not exist either.

$f$ has no one-sided limit at $a$.

Hence $f$ is not a piecewise continuous function with one-sided limits.

$\blacksquare$