Implicit function theorem
The Implicit function theorem is a theorem of calculus. It states that a (multiple-variable) function has many inverses hidden in different neighborhoods of the image. That is to say, the graph of the function can be finitely partitioned so that each piece has an inverse function. It is used by mathematicians for local analysis of functions which cannot be globally inverted. It is considered one of the fundaments of the Calculus and does not require complex analysis for proof.