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Prove that if $f(a)>0$ and $f$ is continuous, then there is a $\delta >0$ such that for all $x$, $|x-a|< \delta$ implies $f(x)>0$.

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  • $\begingroup$ Can you share what you've tried? If you write down the definition of continuity, this is almost trivial. $\endgroup$ – user61527 Jan 28 '14 at 1:19
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Hint: Let $\epsilon=\frac{f(a)}{2}$.

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Take $\epsilon=\frac{f(a)}{2}$ and apply the definition of continuity (in terms of $\epsilon,\delta$) on $a$.

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For $\frac{f(a)}2$, there is a $\delta >0$ such that $ |x-a|< \delta $ then

$$|f(x)-f(a)|<\frac{f(a)}2$$

hence

$$- \frac{f(a)}2\lt f(x)-f(a)<\frac{f(a)}2$$

so $ f(x)\gt \frac{f(a)}2$

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