CDF of a transformation of variables with a flat region I'm new to statistics and I started studying it by myself so I'd like to receive and advice about a reasoning related to a flat region in a transformation.
For the transformation given by ('X' is a random variable while 'x' is an outcome)
\begin{equation}
  Y = \left\{ \begin{array}{llll}
  X-a\text{,}& x \gt a\\
  0\text{,}&   -a \lt x \leq a\\
  X+a\text{,}& x \leq -a
 \end{array}\right.
\end{equation}

the text book I use gives a resulting CDF (F_Y) of
\begin{equation}
  F_Y(y) = \left\{ \begin{array}{llll}
  F_x(y+a)\text{,}& y \gt 0\\
  F_x(a)\text{,}&   y = 0\\
  F_x(y-a)\text{,}& y \lt 0
 \end{array}\right.
\end{equation}
Would the following reasoning be correct regarding the CDF (F_Y) for y = 0?
For y=0, we get 
\begin{equation}
F_Y(y) = P( Y \leq y ) \Rightarrow F_Y(0) = P(Y \leq 0) = P( Y \lt 0 ) + P( Y = 0 )
\end{equation}
According to the transformation given, there is non-zero probability mass concentration at y = 0 given by
\begin{equation}
P( Y = 0 ) = P(-a\leq X \leq a) = P( X \leq  a ) - P( X \lt -a )  
\end{equation}
So, 
\begin{equation}
\begin{array}{111}
F_Y(y=0) =  P( Y < 0 ) + P( X \leq a ) - P( X \lt -a )  
\end{array}
\end{equation}
Also according to the transformation, 
\begin{equation}
P( Y \lt 0 ) = P( X \lt  - a ) 
\end{equation}
Hence, 
\begin{equation}
F_Y(y = 0) =  P( X \lt  - a ) + P(X \leq a ) - P(X \lt -a ) = F_X(a)
\end{equation}.
 A: You can visualize the CDF of $Y$, $F_Y$, in terms of (a) a graph of the transformation $\phi:X\to Y$ and (b) a graph of the CDF of $X$, $F_X$.
In this figure the top plot shows the graph of $\phi$ while the second plot shows the CDF.

To see the value of $F_Y$ at some trial value $y$, draw a horizontal line at height $y$ in the upper plot (shown at $y=3/8$).  Identify all the $x$ for which the graph of $\phi$ lies on or below that line: these are the values of $X$ corresponding to the event $Y\le y$, written $\phi^{-1}(Y \le y)$.  They are shown beneath the red parts of the graph.  In the lower plot, the values of $F_X$ at this event cover a portion of the vertical axis between $0$ and $1$ (as marked in black).  The total amount of coverage is the probability of $\phi^{-1}(Y \le y)$.  It is equal to $F_Y(y)$.
Consider what happens when, as in this example, $y$ is increased slightly so that it suddenly includes a flat portion of the graph of $y$.

Instantaneously, all the probability of $X$ corresponding to that flat portion of the graph of $\phi$ is included within $\phi^{-1}(Y \le y)$.
This visual understanding should give you confidence in your calculations, as well as provide intuition for how distributions behave generally when random variables are transformed--even when the transformations are discontinuous (their graphs have jumps), not one-to-one (they don't always increase or always decrease), or have flat spots.
