For 2., all you need is to write down a function that has period 1 in x and in y. If it was a function of one variable, you could use something like .
The answer to 3. is given by the hairy ball theorem.
Im kinda lost.
2) Let's say I use f(x,y) = cos(2pi xy). When y = 0, the answer would be 1. However if y=1, i would have a variable cos(2pi x). To achieve period 1 i guess the easiest functions to use would be sin and cos, but due to the problem I just described, i dont know how to proceed.
1) I guess here i'd just graph a function similar to the one im trying to find in 2?
3) I guess i'd be possible, but it would never be smooth.