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Suppose $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is a continuous function. Suppose that $F$ attains a local minium in a point $a$. Is true that there exists some ball centered in $a$ such that $F$ restricted to this ball is convex?

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    Interesting, I guess this wasn't as obvious as it seemed in my head. Thanks for pointing it out!2012-11-16

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Investigate the behavior of $x^2 (\sin^2\frac1x + 1)$ at $0$.