2

Let $M$ and $N$ be smooth manifolds and $$f:M\times N\to \mathbb{R}$$ a map. Suppose that the maps $$M\to\mathbb{R},\quad p\mapsto f(p,q_0)$$ $$N\to\mathbb{R},\quad q\mapsto f(p_0,q)$$ are smooth for all $(p_0,q_0)\in M\times N$.

Is $f$ smooth?

Edit: What if $f$ is continuous?

Community
  • 1
user352183
  • 21
  • Since this a a strictly local question, it can be reduced to the case where $M$ and $N$ are open balls in euclidean space. BTW, by "smooth", do you mean $C^\infty$? – John Hughes Jul 05 '16 at 12:32
  • @JohnHughes Yes "smooth" means $C^\infty$. – user352183 Jul 05 '16 at 12:33
  • This looks as if it's going to become a chameleon question, so I'm going to stop answering. – John Hughes Jul 05 '16 at 12:41
  • @JohnHughes Thanks for your answer. It will not become a chameleon question. I am just wondering about continuity. – user352183 Jul 05 '16 at 12:42
  • You might want to look at https://books.google.com/books?id=dULTBwAAQBAJ&pg=PA244&lpg=PA244&dq=if+a+function+on+the+plane+has+smooth+partials+of+all+orders,+is+it+smooth&source=bl&ots=s0gY7Quue2&sig=8KNpfoO00LkV3gImGrpzN5C7sfM&hl=en&sa=X&ved=0ahUKEwimv5byvdzNAhUP3GMKHVsoAYcQ6AEIKzAC#v=onepage&q=if%20a%20function%20on%20the%20plane%20has%20smooth%20partials%20of%20all%20orders%2C%20is%20it%20smooth&f=false which suggests that the answer might be "in that case, it's smooth," although MacLane seems to define smoothness as "all partials exist". He may have continuity as a general assumption, too. – John Hughes Jul 05 '16 at 14:01

1 Answers1

1

See this stackexchange question/answer, which gives a (negative) answer to the simplified question (i.e., on Euclidean space).

(This answer seems to have inspired the edited question in which the assumption of continuity is added, so it's no longer a complete answer, but I'm leaving it because it may prove relevant to someone else.)

Community
  • 1
John Hughes
  • 93,729