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

1. ## 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

2. 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

3. I was hoping to find a non-trivial example, but thank you. Yeah, I agree that Sard's probably isn't applicable here.

4. Oops! I meant critical values, not critical points

5. Solved! Sum of bump functions