Piecewise Continuous Function with Improper Integrals may not be Bounded

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

Let $f$ be a piecewise continuous function with improper integrals.

Then $f$ may not be piecewise continuous and bounded on $\left[{a \,.\,.\, b}\right]$.

Proof
Consider the function:


 * $f \left({x}\right) = \begin{cases}

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

Since $\dfrac 1 {\sqrt{x - a} }$ is continuous on $\left({a \,.\,.\, b}\right)$, $f$ is continuous on $\left({a \,.\,.\, b}\right)$.

Therefore, $f$ satisfies $(1)$ in the requirements of a piecewise continuous function with improper integrals for the subdivision $\left\{ {a, b}\right\}$ of $\left[{a \,.\,.\, b}\right]$.

Also:

Hence $\displaystyle \int_{a+}^{b-} f \left({x}\right) \rd x$ exists.

Thus $f$ is a piecewise continuous function with improper integrals.

However, we have that $f \left({x}\right)$ approaches $\infty$ as $x$ approaches $a$ from above.

Thus $f$ is not bounded.

Therefore $f$ is not piecewise continuous and bounded.

Hence the result.

Also see

 * Bounded Piecewise Continuous Function has Improper Integrals