1. Yes, Option<T> is one of the ways to express "optional" value. And T? is another one. They express the same meaning.

2. No, totality has nothing to do with the expected value. doubler2 in your example is defined for any integer. The fact that it returns null for some of them doesn't make this function partial.

If you say that doubler2 should return doubled input, but it returns null for 6 instead, it just means that this function is implemented incorrectly – it doesn't match the specification. But it's still total.

If you want to demonstrate the idea of a partial function, you can throw an exception for some input, as I suggested in the beginning. Or you can add infinite loop for some input. But null is a valid output from the totality point of view.