In general, $c$ need not be a regular value of $g$ for any $\epsilon>0$. For example $0$ is a regular value of $f(x)=e^x\sin x$, but it approaches to $0$ as $x\to -\infty$ and you can find arbitrarily small perturbations $g$ of $f$ so that $g^{-1}(0)$ will contain an interval. 

PS. I edited your question, because contrary to what you said, $f^{-1}(c)$ need not be compact. Please, think about your question more carefully and ask it as a new one instead of editing this one.