Generalized Implicit Function Theorem and General Fundamental Theorem of Calculus

We present the notion of Henstock-Kurzweil integral for mappings assuming values in Hausdorff topological vector spaces using the direct set of gauges and derive a version of Mean Value Theorem. We use the definition of Frechet derivative and obtain a general version of Implicit Function Theorem for mappings from X \times Y \rightarrow Z where, for existence and continuity of the function, X needs to be merely a topological space and for differentiability, X can be a Topological Vector Space (TVS) while Z is a Hausdorff topological vector space and Y is a Banach space. The implicit function theorem is proved in 3 parts as existence, continuity of the partial derivative and invertibility of the partial derivative. The proof is very similar to the classical proof.