For interval (0 1), Uniform Distribution Function f(x) defined as 1/(1-0) = 1
i.e. f(x) = 1
So function is NOT Uniformly Distributed.
Lets Check Countinuity,
f(x)=e^{-1/x}/x is everywhere continuous except 0. so it will be Continuous in (0 1).
Value of Function at x=0 is Undefined so it should be UnBounded in (0 1)
B should be correct Choice.