is age a discrete or continuous variable
Both. It is correct to consider age as a continuous variable, but we collect data on people's ages as if they were discrete variables. Age is generally given in years, at least in humans. When we round off our numbers, we make age a discrete variable. We say a person turns 21 as if one day he's 20 years old and the next day he's 21 years old. This is the way we treat age as a discrete variable. If we considered age as a continuous variable, we would say he is 20.99 years old a day before his birthday and exactly 21 years old only on his birthday, on the exact time of his birth. After that he is 21.001 years old.