Need help understanding this definition of step functions In my lecture notes, it says:

$\phi [a, b] \rightarrow \mathbb{R}$ 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 $i=1,2,..., k-1, k$

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. 
 A: The second sentence essentially goes into detail on the first part. It says that it divides the interval $[a,b]$ by "cutting it up" at points $p_0<p_1<\dots < p_{k-1}<p_k$.  Since those are the points we cut $[a,b]$ at, we can make a set $P$ serving as a reminder of where we cut:
$$P = \{a=p_0,p_1,p_2, ..., p_{k-1},p_k=b\}$$
note though, it's mandatory that our first cut was at $a \in \mathbb R$ and last cut was at $b \in \mathbb{R}$.
The notation $[a,b]\backslash P$ is another way of writing the set $[a,b]-P$. Being aware that $P \subset [a,b]$ (essentially a collection of points within the interval $[a,b]$), then 
$$[a,b] \backslash P = (p_0=a,p_1) \cup (p_2,p_3) \cup \cdots \cup (p_{k-1},p_k=b)$$
and those are our open sub-intervals.
Lastly, the notation $\phi |_{p_{i-1},p_i}$ means the function we defined before, $\phi$, but instead of $\phi: [a,b] \to \Bbb R$ it's $\phi: (p_{i-1}, p_i) \to \mathbb R$. So now when we plug in points into this function, our domain has to be within $(p_{i-1}, p_i) \in [a,b]$, which we want equal to some $c_i$ (constant) for $i=1,2, \dots , k-1, k$.

Edit: Changed definition of $\phi|_{p_{i-1}p_i}$ based on the response given by Fred.
A: The open subintervalls of $[a,b] \backslash P$ are the intervalls
$(p_0,p_1), (p_1,p_2),...(p_{k-1},p_k)$
I think it should read $\phi |_{(p_{i-1},p_i)}$ instead of $\phi |_{p_{i-1},p_i}$.
$\phi |_{(p_{i-1},p_i)}$ is the resrtiction of $ \phi$ on the intervall $(p_{i-1},p_i)$, hence
$\phi(x)=c_i$ for all $x \in (p_{i-1},p_i)$.
A: Take the interval $[a,b]$ and remove the points $p_k$ (which include the points $a$ and $b$). This splits $[a,b]$ in open subintervals $(p_{i-1},p_i)$, between these points.
$\phi|_{p_{i-1},p_i}$ denotes the restriction of $\phi$ over such a subinterval, and you have a step function when all restrictions are constant functions.

By the way, this definition doesn't say a word about the values at the splitting points, $\phi(p_i)$.
