# R->R function in C∞ whose set of critical points are dense

• Sep 29th 2010, 03:23 PM
ajskim
R->R function in C∞ whose set of critical points are dense
I need to find such a function. Any ideas about how to approach this? We have just covered Sard's theorem if that provides some sort of insight on how I should tackle this.
Thanks
• Sep 29th 2010, 04:14 PM
tonio
Quote:

Originally Posted by ajskim
I need to find such a function. Any ideas about how to approach this? We have just covered Sard's theorem if that provides some sort of insight on how I should tackle this.
Thanks

Sard's theorem is trivial for one variable functions, as \$\displaystyle f(X)=\{0\}\,,\,\,X=\$ the set of critical points of \$\displaystyle f\$ ,

so I don't think it'll help a lot here.

Of course, \$\displaystyle f(x)=c=\$ constant is a trivial example, but I'm not sure how to construct other examples, or even whether there exist...

Tonio
• Sep 29th 2010, 05:17 PM
ajskim
I was hoping to find a non-trivial example, but thank you. Yeah, I agree that Sard's probably isn't applicable here.
• Sep 29th 2010, 05:39 PM
ajskim
Oops! I meant critical values, not critical points
• Sep 29th 2010, 07:44 PM
ajskim
Solved! Sum of bump functions