Integrating with dirac delta I am curious about the dirac delta functions that represent the jumps in the following function.
$$ f(x) = \left\{
     \begin{array}{lr}
       \frac{3}{2} & : x \in (-\infty,-2]\\
       0 & : x \in (-2,-1)\\
       \frac{3}{2} & : x \in [-1,0)\\
       -\frac{3}{2} & : x \in [0,1]\\
       0 & : x\in(1,\infty)
     \end{array}
   \right.$$
Would the jump at $-2$ be represented by $\frac{3}{2}\delta(x+2)$? Would it be positive or negative? In other words, do we see the jump as going down from $\frac{3}{2}$ to $0$ or up from $0$ to $\frac{3}{2}$?
How would we view the jump from $\frac{3}{2}$ to $-\frac{3}{2}$ at $0$?
 A: What you're really after is the step function, $$H(x) =
\begin{cases}
 0 & x<0 \\\\
 1 & x\geq0.
\end{cases}$$
In which case you could write $g(x)=\frac32H(-2-x)-\frac32H(x-1)$ for $$g(x) =
\begin{cases}
 \frac32  & x\leq-2 \\\\
 0  & 2<x<1 \\\\
 -\frac32 & 1\leq x,
\end{cases}$$
for example. Here are two ways of thinking about the step function $H$.
As hinted in your original question, the distributional derivative of the Heaviside step function is the delta function. That is, $H'(x)=\delta(x)$. This is a way to formalize the geometric "jumping" intuition you observed. So while your original function $f$ is not a linear combination of delta functions, it's distributional derivative is.
A much more widely-used concept is that of an indicator or characteristic function. Considering $A=[0,\infty)$ be a subset of $\mathbb{R}$, then we can write $\chi_A(x)=H(x)$. In general, we can let $A$ be a subset of any set $X$ and define, $$\chi_A(x) =
\begin{cases}
 1 & x \in A \\\\
 0 & x \notin A.
\end{cases}$$
Basically, this function answers the question, "is $x$ an element of the subset $A$ of $X$?" So we could write the function $g$ as $\frac32\chi_A-\frac32\chi_B$ where $A=(-\infty,-2]$ and $B=[1,\infty)$ are subsets of $\mathbb{R}$.
Hopefully this discussion gives you a few ways of thinking about your original function $f$.
