Definition: If $f$ is a simple function with range $R(f)=\{a_1,a_2,\dots,a_n\}$ and $E_k=\{x\in D(f):f(x)=a_k\}$ for $k\in \{1,2,...,n\}$ then the Canonical Representation of $f$ is $$\begin{align} \quad f(x) &= \sum_{k=1}^{n} a_k \chi_{E_k}(x) \\ &= a_1\chi_{E_1}(x) + a_2\chi_{E_2}(x) + ... + a_n\chi_{E_n}(x) \end{align}.$$
If each term in the sum is a constant $\times$ a characteristic function $\{0,1\},$ is the sum a constant? The confusion comes from picturing each term as a separate step in things like this:
If the function is composed of all the different steps in red segments in the plot, then adding arithmetically the different constant values doesn't seem to represent the function well, because you end up with just a single constant - not multiple constant values, one for each step. On the other hand, if each step is a separate simple function, then the sum doesn't make sense.
My problem may lie in understanding what is involved in the operation $a_k\chi_{E_k}(x).$ Is this sum an arithmetic sum, or is the $\sum$ symbol simply a representation of a sequence of simple functions in this context?
No. The function is not constant, since there are different points in which it takes different values. At most one term in the sum is not zero, since the $E_k$ are disjoint. The sum is equivalent to the following: $$ f(x)=\begin{cases} a_1 & \text{if }x\in E_1\\ a_2 & \text{if }x\in E_2\\ \dots &\\ a_n & \text{if }x\in E_n\\ 0 & \text{if }x\notin\bigcup_{k=1}^nE_k \end{cases} $$