If by Birth Rate you mean the number of births during a given period of time then the answer is yes. The rate may be further qualified, for example, as the number per 1000 women of childbearing age but that does not alter the nature of the variable.
If, however, by birth rate you mean the number of births WITHOUT the time element, the measure becomes discrete.
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Continuous.Continuous.Continuous.Continuous.
It is a continuous variable.
Yes. It is a continuous variable. As used in probability theory, it is an example of a continuous random variable.
No, Patient's gender is not a continuous variable. It is discrete variable.
No, it is a continuous process but not a continuous variable. The magnitude of star-shine, across all stars is a continuous variable. The magnitude of a star's shine over time is a continuous variable.