In my lecture notes, it says:

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> $\phi [a, b] \rightarrow \mathbb{R}$ is a *step function* if there exists a finite set $P \subset [a,b]$ with $a, b \in P$ such that $\phi$ is constant on the open sub-intervals of $[a,b] \backslash P$. $P$ is called a *partition* of $[a,b]$. More concretely, $P = \{a=p\_0,p\_1,p\_2, ..., p\_{k-1},p\_k=b\}$, with $p\_0 < p\_1 < p\_2 <...<p\_{k-1} < p\_k$ and $\phi |\_{p\_{i-1},p\_i} = c\_i$ (a constant), for $i=1,2,..., k-1, k$
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There are a few details I don't understand about this definition. I understand the general idea of what a step function is, but I need help on understanding 

* what does it mean when it says "the open sub-intervals of $[a,b] \backslash P$"?
* everything following "more concretely", as I don't understand it and, in particular, I am not sure what $\phi |\_{p\_{i-1},p\_i}$ means.

Thanks for the help.