How to sketch output PDF given the transformation? Question:
Let Random variable $X$ with $PDF$ = $f_{X}(x)$ be the input to device with input output characteristics as shown below then sketch the $PDF$ of $Y$ i.e, $f_{Y}(y)$





My attempt:

for  X>0

,$Y=X+1\implies  f_{Y}(y)=f_{X}(y-1)$

for X&lt;0

,$Y=X-1\implies  f_{Y}(y)=f_{X}(y+1)$

but at 

X=0 there is $2$ units jump from $-1$ to $1$ and derivative of jump discontinuity is impulse so i thought there should be presence of impulse in PDF of Y but i'm not sure about it .

any help in sketching O/P PDF Y  will be greatly appreciated, and it is not a homework problem but from local author textbook
There is no impulse because there is zero probability mass at $0$. As the problem itself shows, it is perfectly all right for a pdf to make sudden jumps in value; note that the graph of $f_X$ (yes, I do mean $X$, not  $Y$) jumps in value at $0$ and the value of $f_X(0)$ cannot be determined from the graph. $f_X(0)$ could have value $\frac 14$ or $\frac 12$ or any value in between (or any nonnegative value for that matter)