Answer: Continuous if looking for exact age, discrete if going by number of years.
If a data set is continuous, then the associated random variable could take on any value within the range. As an example, suppose that the random variable X, representing your exact age in years, could take on any value between 0 and 122.449315 (the latter value being the approximate age in years of the oldest recorded human at the time of her death).
Under this method, #5#, #5.1#, #5.01#, #5.0000000000000000001#, etc, would all be distinct ages. We would have an infinite number of values within a finite range. As a result, the probability that the random variable would take on any particular one of these infinite values is said to be zero. (However, one could determine the probability of the variable taking on a value less than or greater than a given reference value).
Typically, however, age is truncated; a person who is #5.1# years old and a person who is #5.6# years old would both be said to be #5# years old. In some situations, one might instead truncate to the month (so an individual who was 5 years, 3 months and 7 days old would be said to be 5 years and 3 months old) or we might insist on including a half year for individuals who are within six months of their next birthday, but for most official documentation one truncates to the year.
In such a case, #Y#, representing the age of an individual in years, could take on a finite number of values within our range. #Y# could be 9, or 42, or 75, but it couldn't be 75.5, for example. The data distribution for this random variable would be discrete. As a result, the probability of the random variable taking on any of these finite discrete values within our range could be non-zero.