  This article presents an elementary proof of the Implicit Function Theorem
for differentiable maps F(x,y), defined on a finite-dimensional Euclidean
space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base
point. In the case of a single scalar equation this continuity hypothesis is
not required. The Inverse Function Theorem is also shown. The proofs rely on
the mean-value and the intermediate-value theorems and Darboux's property (the
intermediate-value property for derivatives). These proofs avoid compactness
arguments, fixed-point theorems, and integration theory.
