When the piecewise constant integral independs of the partition's choice?

Question

Proposition

Let $I$ be a bounded interval, and let $f:I\to\textbf{R}$ be a function. Suppose that $\textbf{P}$ and $\textbf{P}'$ are partitions of $I$ such that $f$ is piecewise constant both with respect to $\textbf{P}$ and with respect to $\textbf{P}'$. Then \begin{align*} p.c.\int_{[\textbf{P}]}f = p.c.\int_{[\textbf{P}']}f \end{align*}

My question

Although my question is quite elementary, I would like to ask the following: if $f$ is piecewise constant with respect to $\textbf{P}$ and $\textbf{P}'$, then necessarily $\textbf{P}$ is finer than $\textbf{P}'$ or $\textbf{P}'$ is finer than $\textbf{P}$?

Answer 1

No. For instance, consider a constant function and any two partitions.