Suppose is a function such that . What conditions on will guarantee that the equation can be solved for as a function of near .

So obviously is one condition just by what the hypothesis of the implicit function theorem uses (I think this part is obvious, but I could be wrong and if I am, please explain why this is part of a condition that will guarantee what we want). However, another condition that is needed that will guarantee what we want is , and I just don't see how to arrive at that conclusion. Any help would be appreciated.