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Suppose I have a function of $x$, depending on a small parameter $\epsilon$. I write the function as $f(x, \epsilon)$ where $\epsilon$ is small. For simplicity let $x \in \mathbb{R}$ and $f(x, \epsilon) \in \mathbb{R}$.

Assume that $f(x,0)$ is nonzero for all $x$. So either $f(x,0)<0$ or $f(x,0)>0$. Can I somehow make use of implicit function theorem that for all small $\epsilon$, I would also have $f(x, \epsilon) \neq 0$ for all $x$?

If so, how would I explicitly set it up to make use of the implicit function theorem?

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    I don't think this works, see $x,\epsilon\mapsto (-1)^{x\epsilon}$. – 2017-01-16
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    What if I place some assumptions on derivatives of $f$? @YoTengoUnLCD – 2017-01-16
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    To be honest I don't understand your example, as $(-1)^{x\epsilon}$ is nonzero and is not even well defined... @YoTengoUnLCD – 2017-01-16

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