What's an example of a Riemann integrable, non-regulated function?
Definitions:
Let $X$ be a normed space and $[a,b]$ be a compact of $\mathbb R$.
Step Functions:
A function $f: [a,b] \longrightarrow X$ is said to be a step function if: there exists a partition $P=\{x_0 = a, x_1, \ldots, x_n = b \}$ of $[a,b]$, such that $f$ is constant on each subinterval $(x_i, x_{i+1})$.
Riemann integrable Functions:
A function $f: [a,b] \longrightarrow X$ is said to be integrable if for any $\epsilon > 0$, there are two step functions $\Phi: [a,b] \longrightarrow X$, and $\theta: [a,b] \longrightarrow \mathbb R_+$, such that:
$$\begin{cases}
||f(x) – \Phi(x)|| \le \theta(x), \ \forall \ x \in [a,b] \\
\int_a^b \theta(x)dx \le \epsilon
\end{cases}$$
Regulated Functions:
$f: [a,b] \longrightarrow X$ is said to be regulated if:
$\forall \ \epsilon > 0, \ \exists \ \Phi: [a,b] \longrightarrow X$, a step function, such that: $||f(x) – \Phi(x)|| < \epsilon$, $\forall$ $x \in [a,b]$.
A regulated function is Riemann integrable: for $\epsilon > 0$, there is a step $\Phi: [a,b] \longrightarrow X$, such that: $||f(x) – \Phi(x)|| < \epsilon/4(b – a)$. We then take: $\theta(x) = \epsilon/2(b-a)$. So that $(\Phi, \theta)$ is a pair of step functions associated to $f$.
What about the converse? Need a Riemann integrable function be regulated? I am thinking not. But I'm not finding a way to prove it.
Thank you.
Best Answer
Hint: Regulated function has left and right limit at every point.