Question

Is discrete random value can take any value within an interval on the real line?

Answer 1

In case of discrete random, variable, the probability can either be "zero" or "one" only and no other values other  than this is there.

On the other hand, we we attempt to add up these, the sum total will come up to "one".

On the other hand still, in case of a continuous random variable, the scenario is such that there is an acceptance of any value i.e. numerical  taken in between intervals like in case of playing with the ludo's dice.

This is an important part  of statistics.

Again, in case of continuous random, variable one can take up any value that one likes and this will go on.