# Can a continuous map on a Hilbert manifold be approximated by a map which has infinitely many critical points?

It is well-known that a continuous map $f:M\to\mathbb{R}^n$ from a Hilbert manifold can be closely approximated by a smooth map $g:M\to\mathbb{R}^n$ which has no critical points.

But, can such a continuous map $f$ also be closely approximated by a map $h:M\to\mathbb{R}^n$ which has infinitely many critical points? Also, are these critical points dense in $M$?

A reference would be much appreciated. Thanks in advance! Cross-posted on MSE.

Edit: As mentioned by Pietro, a $C^0$ perturbation creates infinitely many local minima and maxima. However, do we rigorously show this?

• No difficulty at all in creating infinitely many local minima and maxima by a small $C^0$ perturbation. Of course impossible, in general, by a small $C^1$ perturbation. – Pietro Majer Aug 3 '18 at 19:30
• @PietroMajer Would you mind explaining why a $C^0$ perturbation creates infinitely many local minima and maxima? Could you also add this as an answer, if you don't mind? – Sergio Charles Aug 3 '18 at 23:39
• @Multivariablecalculus: Just take the function that you want to approximate itself, but in a small neighborhood make the function constant using a partition of unity. The neighborhood can be made arbitrarly small. – Thomas Rot Aug 4 '18 at 14:48
• @ThomasRot Do you mind including this as an answer, for the sake of completeness? – Sergio Charles Aug 4 '18 at 16:04
• Working in local charts. Let $\sigma(t):=\max\big(0,\min(2t-1,1)\big)$ for $t\in\mathbb{R}$. For $\epsilon>0$ define $f_\epsilon(x):=f\big(\sigma (\|x\|/\epsilon)\ x\big)$. Then $f_\epsilon$ is constant in the ball of radius $\epsilon/2$, coincides with $f$ outside the ball of radius $\epsilon$, $\|f-f_\epsilon\|_\infty=o(1)$ for $\epsilon\to0$ – Pietro Majer Aug 5 '18 at 17:03

As suggested by Pietro, a continuous map $f:X\to\mathbb{R}^n$ on a Hilbert manifold $X$ may be approximated to have infinitely many points. That is, a a $C^0$-perturbation creates infinitely many local minima and maxima. However, this is, in general, not true for a $C^1$-perturbation. In particular, in local charts, suppose $\sigma(t):=\max(0,\min(2t-1,1))$ for $t\in\mathbb{R}$. Then for $\varepsilon>0$, define $f_{\epsilon}(x):=f(\sigma(\|x\|/\varepsilon)x)$. As such, $f_{\varepsilon}$ is constant in the ball of radius $\epsilon/2$, it coincides outside the ball of radius $\varepsilon$, and $\|f-f_{\varepsilon}\|_{\infty}=o(1)$ for $\epsilon\to 0$.