TransformedRegion with piecewise function

Is it possible to use TransformedRegion with a Piecewise defined function?

I tried the following, but it doesn't work and I don't understand why.

V[{x_, y_}] = Piecewise[{{{2 x , y}, x < .5 }, {{.5 x , y}, x >=  .5}}];
Region[
TransformedRegion[Rectangle[], V]
]

EDIT: How does TranformedRegion transform a region? Why does it not work with simple discontinuous or non-differentiable transformations?

• What happens if you use Region[TransformedRegion[Rectangle[], {Piecewise[{{2 Indexed[#, 1], Indexed[#, 1] < 1/2}, {Indexed[#, 1]/2, Indexed[#, 1] >= 1/2}}], Indexed[#, 2]} &]]? May 2 '20 at 11:14

The map under consideration is discontinuous on Rectangle[] . I have doubts whether TransformedRegion works with such maps. The command

Region[TransformedRegion[Rectangle[], Function[p, {Piecewise[{{2*p[], p[] < 1/2},
{1/2*p[], p[] >= 1/2}}], p[]}]]]

crashes the kernel on my comp and the command

Region[TransformedRegion[Rectangle[], Function[p, {Piecewise[{{2*p[], p[] < 1/2},

{3*p[] - 1/2,p[] >= 1/2}}], p[]}]]]

, where the map is continuous, works well.