will a sharp knife which is infinitely long and touching every point of the surfave be able to cut through the

Question

surface because then the area will be infinite.

Answer 1

Ouch!
Infinitely long knife requires an infinite force since to cut any infinitely long surface, no matter how realistically sharp the knife is (and hopefully it is not infinitely sharp) it will require some force per unit of length. So if your force is a bit shy of infinit it will not cut the surface.

When you say it touches ever point on the surface then a knife is infinitely dull.

Answer 2

No. If it's touching every point of the surface, it's already inside it (draw yourself a picture)

Answer 3

Yes, of course. I don't think it should be explained, because it's too easy.