Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Define a smooth function $f(x,y)$ on $\mathbb{R}^2$,which satisfies the following condition:$$f(x,y)=f(x+1,y);\quad f(x,y)=f(x,y+1)$$ (i.e.$f$ can be defined on a torus)then hotw to proof that there are at least 4 critical points($f_{x}=f_{y}=0$) on the unit cube $[0,1]\times[0,1]$ (obviously,it has at least 2 critical point)?

I'm also like to know other examples of thhis kind(the topology implies some properties of the function defined on that space)

*Edit*I think there are 2 different proffs of this question,one only need some use of calculus techniques(which I tried but failed),another one may include some topological statements like Neal said.

share|improve this question
First, what have you tried? Second, what theory have you established so far? For example, do you know the Poincare-Hopf index theorem? Or do you know properties of Morse functions? –  Neal Sep 20 '12 at 12:51
(Also, you might be more likely to get a response if you raise your accept rate.) –  Neal Sep 20 '12 at 13:04
I tried to use some basic calculas to prove it(construc some auxiliary function)but failed.BTW,I don't have much backgroud on topology,but I'm very glad if you can show me that way –  sun Sep 20 '12 at 13:14
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.