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May 1, 2019 at 19:53 comment added Onil90 Oh, right! Thanks a lot for the clarification!!
May 1, 2019 at 19:52 vote accept Onil90
May 1, 2019 at 17:15 comment added RBega2 @Onil90 Yes. Think of $f_\epsilon(x)=x^4-2 \epsilon^2x^2$. This has strict local minima at $x=\pm \epsilon$ and a strict local maxima at $x=0$ when $\epsilon>0$. However, $f_0=x^4$ has only a degenerate local minima at $x=0$.
May 1, 2019 at 14:31 comment added Onil90 Thank you for your answer! So essentially you are saying that the limit metric $g$ in my example cannot actually be the standard metric of $\mathbb{S}^2$?
May 1, 2019 at 12:14 history edited RBega2 CC BY-SA 4.0
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May 1, 2019 at 1:01 history answered RBega2 CC BY-SA 4.0