set $\Omega$ or of the noncommutative function $f$.

A remarkable result (see [@ncfound Theorem 7.2]) states that, under very mild conditions on $\Omega$, local boundedness for $f$ implies each $f_n$ is analytic as a map between Banach spaces. Indeed, a hint towards the proof of this result is the following essential property of noncommutative functions: if $\Omega$ is admissible, $a\in \Omega_n, c\in\Omega_m, b\in M_{n\times m}
(\mathcal A)$, such that $\left(\begin{array}{cc}
a & b \\
0 & c
\end{array}\right)\in\Omega_{n+m}$, then there exists a linear map $\Delta f_{n,m}(a,c)\colon M_{n\times m}(\mathcal A)\to M_{n\times m}
(\mathcal C)$ such that $$\label{FDQ}
f_{n+m}\left(\begin{array}{cc}
a & b \\
0 & c
\end{array}\right)=\left(\begin{array}{cc}
f_n(a) & \Delta f_{n,m}(a,c)(b) \\
0 & f_m(c)
\end{array}\right).$$ Obviously, this implies in particular that $f_{n+m}$ extends to the set of all elements $\left(\begin{array}{cc}
a & b \\
0 & c
\end{array}\right)$ such that $a\in \Omega_n, c\in\Omega_m, 
b\in M_{n\times m}(\mathcal A)$ (see [@ncfound Section 2.2]). Two properties of this operator that are important for us are $$\label{FDC}
\Delta f_{n,n}(a,c)(a-c)=f(a)-f(c)=\Delta f_{n,n}(c,a)(a-c),\quad
 \Delta f_{n,n}(a,a)(b)=f_n'(a)(b),$$ the classical Frechet derivative of $f_n$ in $a$ aplied to the element $b\in M_n(\mathcal A)$. Moreover, $\Delta f_{n,m}(a,c)$ as functions of $a$ and $c$, respectively, satisfy properties similar to the ones described in 