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In the second derivative test for extrema is it necessary that the second derivative must be continuous at the concerned point or it must preserve its sign in some neighborhood of the concerned point, because in many books I have seen that it is given that if $f$ is a twice differentiable function and let $c$ be a critical point such that $f'(c)=0$ and $f''(c)>0$ then the function has a minima.

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You only need existence of $f''(c)$ and the fact that $f''(c)>0$. Nothing about existence of second derivative in a neighborhood of $c$ is needed.

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