An integer-valued function is a type of mathematical function where the output can only be a whole number or "integer". For example, if you took the number 7 and doubled it, the answer would be 14. The result can't be 7.5 or any other fraction, it has to be a whole number.
Another example of an integer-valued function is when you take a number and divide it by 2. In this case, the answer has to be a whole number too. If you take 6 and divide it by 2, you can't get 3.5 as the answer - it has to be 3.