I'm delving into the differentiability of composite functions and seeking clarity on a specific conceptual approach. Suppose we have a function (g) that is known to be differentiable on a set (X), and another function (f) that is differentiable on a set (U). Let's consider (Y) as the range of (g) when applied to (X).
My question revolves around the intersection of (Y) and (U): If we look at the region (Y ∩U), can we assert that the composite function (f(g(x))) is differentiable in this intersection? In other words, does the intersection of the derivable ranges of (g) and (f) provide a reliable indication of the composite function's differentiability?
Any insights or explanations on this matter would be greatly appreciated.