How to define inclusions in Mathematica?

Questions
1

Dear members of the Wolfram Community,
Does any one has define inclusions (set valued maps) such as
y=0 for x>0 and y=(-inf,0] for x=0, in Mathematica??
Thanks

Jesus Rico

2 
y = Piecewise[{{0, x > 0}, {Interval[{-\[Infinity], 0}], x == 0}}]
3

Thanks!
I used your definition within a Plot function

Plot[y, {x, -1, 1}, Exclusions -> None]

and get only a horizontal line that not correspond to expected function.

What am I doing wrong?

Jesus

4

It’s not really a function. It has an infinite number of values of y for x = 0.
I don’t think Plot can handle that.

5

Differential inclusions are very interesting generalizations of ordinary differential equations and definitions like the one you help me with are needed.

Thanks indeed!

Jesus