Inverse and implicit function theorems

Let F be continuously differentiable on an open subset of real n-space, and suppose its derivative at p is an invertible n-by-n matrix. There are open neighborhoods of p and F(p) on which F is a continuously differentiable diffeomorphism. If F is smooth, its local inverse is smooth. The inverse derivative is obtained by differentiating the identity, not by taking coordinatewise reciprocals of Jacobian entries. Continuity of the derivative is a convenient sufficient regularity assumption that controls its nearby behavior.

For maps of manifolds of equal dimension, use source and target charts and test whether the intrinsic tangent map is an isomorphism. Compatible chart changes preserve this condition. The result says nothing about global injectivity, the size of the inverse neighborhood, or how well conditioned numerical inversion is. A determinant can be nonzero but extremely small, making the local inverse sensitive. Conversely, failure of this sufficient test does not imply that the map has no set-theoretic inverse, only that this smooth-inverse conclusion is unavailable.

Suppose G maps an open subset of real (m+k)-space to real k-space, is continuously differentiable, and vanishes at (a,b). Split variables into x and y, with y having k coordinates. If the k-by-k derivative with respect to y is invertible at that point, nearby solutions are exactly y=g(x) for one continuously differentiable function. The theorem uses a chosen block; another block may work when this one fails. Differentiating G(x,g(x))=0 supplies the graph derivative.

This is a coordinate statement about a local level set, not a license to select one formula globally. A circle has upper and lower branches over much of the horizontal axis, and near its left and right extremes it is better expressed with the horizontal coordinate dependent on the vertical one. The geometry can stay smooth while the original graph representation breaks down. Listing which variables are solved for and where their derivative block is invertible prevents a coordinate limitation from being confused with a singular manifold.

The inverse theorem can also complete selected independent coordinates to a full local coordinate system. This leads to normal forms for constant-rank maps and the regular-value theorem. A full-rank equation system cuts out the expected number of independent directions because coordinates can straighten those equations into projections. The difficult part is not merely counting equations but verifying that their differentials are independent at every relevant point of the level set.

Boundary points need separate treatment. An ordinary Euclidean local inverse theorem uses open neighborhoods, whereas a manifold boundary uses half-space charts and imposes compatibility with the boundary. Later preimage results will require the restriction to the boundary to satisfy its own rank or transversality condition. Until then, use the theorems here at interior points. Keep the analytic regularity, derivative block, coordinate domain, and locality visible when transferring a formula into a geometric claim.