Hi, I'm stuck on the following question. Any help will be fantastic, thanks.

Question:

$\displaystyle f:\Re \to \Re $ is a continuous function. Prove that if for some $\displaystyle c \in \Re $, $\displaystyle f(c)>0 $, then there exists a $\displaystyle \delta > 0 $ such that $\displaystyle \forall x \in ( c - \delta, c + \delta) $, $\displaystyle f(x) > 0 $

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I tried a contradiction proof but didn't really get anywhere. Please help! Thank you.