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. 



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> 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 $M$?
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A reference would be much appreciated. Thanks in advance! [Cross-posted on MSE](https://math.stackexchange.com/questions/2870825/can-a-continuous-map-on-a-hilbert-manifold-be-approximated-by-a-map-which-has-in).




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



Can a continuous map on a Hilbert manifold be approximated by a map which has infinitely many critical points?
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 $\epsilon\to 0$.


