A few remarks concerning complex-analytic metric spaces

Let E be a closed set in C, normally with empty interior, and let us consider continuously-differentiable functions on E in the sense of Whitney, in which the differential of the function is automatically included. This means a continuous complex-valued function f(z) on E together with a real-linear mapping dfz : C n → C defined and continuous for z ∈ E, with properties like those of an ordinary continuously-differentiable function, in terms of local behavior. Whitney’s extension theorem states that there is a continuously-differentiable function on C in the usual sense which agrees with f on E and whose differential is equal to dfz when z ∈ E. There are analogous results for stronger smoothness properties, including C conditions. If E is contained in a smooth real submanifold of C, then only the restriction of dfz to the tangent space of the submanifold is determined by f on E. However, the rest of the differential is still relevant for continuouslydifferentiable extensions to C. On a fractal set, it may be that dfz is determined in some or all directions by f on E, even though E is not at all like a manifold in those directions. This happens already for self-similar Cantor sets, for instance. Every real-linear mapping from C into C can be expressed in a unique way as the sum of a complex-linear mapping and a conjugate-linear mapping. For the differential dfz of a continuously-differentiable function f , these are typically denoted ∂fz and ∂fz. The condition ∂fz = 0 for every z ∈ E is a version of holomorphicity for continuously-differentiable functions on E, which is equivalent to the requirement that dfz : C n → C be complex-linear for each z ∈ E. Of course, the restriction to E of a holomorphic function on a neighborhood of E has this property. It is already sufficient to have a holomorphic extension in some directions, in such a way that dfz is the limit of complex-linear mappings for each z ∈ E. If E is a complex submanifold of C, then this condition implies that f is holomorphic on E in the usual sense. If E is totally disconnected, then there are plenty of nonconstant locally constant functions f on E, for which one can take dfz = 0 for each z ∈ E. There are also connected snowflake sets E with a lot of nonconstant continuously-differentiable functions with dfz = 0 for every z ∈ E.