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This article presents an elementary proof of the Implicit Function Theoremfor differentiable maps F(x,y), defined on a finite-dimensional Euclideanspace, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the basepoint. In the case of a single scalar equation this continuity hypothesis isnot required. The Inverse Function Theorem is also shown. The proofs rely onthe mean-value and the intermediate-value theorems and Darboux's property (theintermediate-value property for derivatives). These proofs avoid compactnessarguments, fixed-point theorems, and integration theory.

The Implicit Function Theorem when the matrix $\frac{\partial F}{\partial y}(x,y)$ is only continuous at the base point