The full question is:

Consider the following random experiment: Independently, choose two random numbers X,Y uniformly form [0,1]. Let \(Z := YI_{\{X \leq \frac{1}{2}\}} \)

b) Show that Z is neither a discrete random variable nor a continuous random variable.

My attempt: I can see that Z is continuous if \(X \leq \frac{1}{2} \) and discrete if \( X < \frac{1}{2} \) but how do we show that it is neither?

Also I can show that If we calculate the expected value of Z assuming it were discrete, then \(\mathbb{E}Z = 0\) but I dont know how to go about the continuous case.