Are all infinite random variables necessary continuous?Explain.

Are all infinite random variables necessary continuous?Explain.
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The rule or method that is used to assign a particular value to a variable which is the outcome in the sample space of an event is known as Random variables.
These random variables are of two types: discrete random variables and continuous random variables.
Discrete variables contain number of finite values whereas continuous variables includes infinitely many values within an interval.
For example,
Consider an infinite random variable X which is the number of times a die is roll until 6 facing up, then the possible values are infinite but not continuous.
Thus, all infinite random variables are not necessarily continuous.
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