On connectedness of derivative sets (Q1332006)

Let \(D\) be a domain in \(\mathbb{R}^ p\), \(f: D\to \mathbb{R}^ 1\) a continuous real function. Denote a point \(x\in D\) by \((x_ 1,\dots,x_ i,\dots,x_ p)= (x_ i,y)\). A number \(d\) is called a partial derivative number (with respect to \(x_ i\)) of \(f\) at the point \(x\) if there exists a sequence of real numbers \(\{h_ n\}\), \(h_ n\to 0\), for which \[ \lim_{n\to 0} {1\over h_ n} (f(x_ i+ h_ n, y)- f(x,y))= d. \] Denote \({\mathfrak m}^{(i)}_ x(f)\) the set of all partial derivative numbers of \(f\) with respect to \(x_ i\) at \(x\). Then the set \[ W= \{(x,\xi): x\in D,\;\xi\in {\mathfrak m}^{(i)}_ x(f)\} \] is a connected subset of \(\mathbb{R}^ p\times \mathbb{R}^ 1\). As a corollary we obtain: If the derivative \({\partial f\over \partial x_ i}\) is finite for any \(x\in D\) then the set \[ W= \left\{(x,\xi): x\in D,\;\xi= {\partial f\over \partial x_ i} (x)\right\} \] is connected. Let \(D\) be a domain of the complex plane \(\mathbb{C}\) and \(f: D\to \mathbb{C}\) a continuous function in \(D\). A number \(d\) is called a derivative number of \(f\) at a point \(z\) if there exists a sequence of complex numbers \(\{h_ n\}\), \(h_ n\to 0\), such that \[ \lim_{n\to\infty} {1\over h_ n} (f(z+ h_ n)- f(z))= d. \] Denote by \({\mathfrak M}_ z(f)\) the set of all derivative numbers of \(f\) at \(z\). Then the set \[ W= \bigl\{(z,\zeta): z\in D,\;\zeta\in {\mathfrak M}_ z(f)\bigr\} \] is a connected subset of \(\mathbb{C}^ 2\).