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The implicit function theorem for continuous functions. (English) Zbl 1173.58004
This paper concerns results of the type of the implicit function theorem and of the Darboux theorem for continuous maps between topological spaces and for differentiable (but not necessarily continuously differentiable) maps between finite dimensional manifolds. While in the authors’ paper [Semigroup Forum 72, No. 3, 353–361 (2006; Zbl 1098.58008)] the existence of a local data-to-solution-map and its continuity inone point is stated, now its continuity in a neighbourhood of this point is proved. For that, homotopy and degree arguments are used.
Unfortunately no applications are presented, in particular no application relevant models where differentiability, but not continuous differentiability appears. Also, the – important for applications – question of local uniqueness, is not touched.

MSC:
58C15 Implicit function theorems; global Newton methods on manifolds
47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators
47H11 Degree theory for nonlinear operators
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