Let $X$ and $Y$ be random variables with a joint PDF $f_{X,Y}$ and suppose $f_{X,Y}$ has a bounded range, i.e., $$\max_{(x,y) \in \mathbb{R}^2} f_{X,Y}(x,y) \leq \kappa$$ for some $\kappa \in \mathbb{R}$.
Let $Z = |X-Y|$ and let $f_Z$ be its PDF. Does $f_Z$ has a bounded range as well?
No. For example, suppose $Z,W$ have joint density
$$f_{Z,W}(z,w) = \cases{\exp(-w^2 z)/\sqrt{\pi} & for $0 < z < 1,\ w > 0$\cr 0 & otherwise}$$
which is bounded. The density of $Z$ is
$$ \int_0^\infty f_{Z,W}(z,w)\; dw = \cases{1/\left(2\sqrt{z}\right) & for $0 < z < 1$\cr 0 & otherwise}$$ which is unbounded.
$Z = X - Y$ where $X =(W + Z)/2$, $Y = (W - Z)/2$, and the joint density of $X$ and $Y$ is related to that of $Z$ and $W$ by a linear transformation, thus is also bounded:
$$ f_{X,Y}(x,y) = \cases{ 2 \exp(-(x+y)^2(x-y))/\sqrt{\pi} & for $0 < x - y < 1$, $x + y > 0$\cr 0 & otherwise}$$
