Yes, most people will give their age as an integer. Even though young children, are more precise and use phrases like "four and a quarter" or "three years and ten months", there are still a small finite number of possible answers.
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Yes, is a discrete variable.
Either fog occurs on a day, or it does not. Therefore it is a discrete value.
Discrete. You cannot have 1.25 homicides!
Discrete. You cannot have 3.76... rooms!
The answer depends on whether or not age is expressed in whole years (discrete) or calculated from the date of birth (near continuous). It also depends on whether the number of observations is such that the data are presented as a grouped two-way table.