\lim_{\stackrel{ z\longrightarrow\alpha}{{\sphericalangle}}}$ the nontangential limit at a point $\alpha\in\mathbb R$ (see, for ex. [@garnett]).

\[JC\] Let $f\colon\mathbb C^+\to\mathbb C^+$ be a nonconstant analytic function and $\alpha\in
\mathbb R$ be fixed.

1.  Assume that $$\label{3}
    c:=\liminf_{z\to\alpha}\frac{\Im f(z)}{\Im z}<\infty.$$ Then $f(\alpha):=\displaystyle\lim_{\stackrel{ z\longrightarrow\alpha}{{
    \sphericalangle}}}f(z)$ exists and belongs to $\mathbb R$, and $$\label{4}
    \lim_{\stackrel{ z\longrightarrow\alpha}{{
    \sphericalangle}}}f'(z)=
    \lim_{\stackrel{ z\longrightarrow\alpha}{{
    \sphericalangle}}}\frac{f(z)-f(\alpha)}{z-\alpha}=
    \lim_{y\downarrow0}\frac{\Im f(\alpha+iy)}{y}=c.$$

2.  Assume that $\displaystyle\lim_{y\downarrow0}f(\alpha+iy)=:
    f(\alpha)$ exists and belongs to $\mathbb R$. If $$\lim_{\stackrel{ z\longrightarrow\alpha}{{
    \sphericalangle}}}\frac{f(z)-f(\alpha)}{z-\alpha}=c\in\mathbb C,$$ then $c\in(0,+\infty)$ and $$c=\liminf_{z\to\alpha}\frac{\Im f(z)}{\Im z}=
    \lim_{\stackrel{ z\longrightarrow\alpha}{{
    \sphericalangle}}}f'(z).$$

3.  Assume that $\displaystyle\lim_{\stackrel{z\longrightarrow
    \alpha}{{\sphericalangle}}}f'(z)=c\in\mathbb C$ and $\displaystyle\lim_{\stackrel{ z\longrightarrow\alpha}{{
    \sphericalangle}}}f(z)=f(\alpha)\in\mathbb R$. Then $$c=\liminf_{z\to\alpha}\frac{\Im f(z)}{\Im z}=
    \lim_{\stackrel{ z\longrightarrow\alpha}{{
    \sphericalangle}}}\frac{f(z)-f(\alpha)}{z-\alpha}\in\mathbb R.$$

The noncommutative version of this theorem becomes quite obvious in light of and of the formulations of the corresponding main result from [@Wlo] as well as the recent work [@PTD]. In the following, when we make a statement about a completely positive map, we usually write the statement for level one, and, unless the contrary is explicitly stated, we mean that the property in question holds for all levels $n$. Thus, for example, the statement $$\lim_{\stackrel{ z\longrightarrow0}{{
\sphericalangle}}}f'(\alpha+zv):=f'(\alpha)$$ exists and is completely positive for $\alpha=\alpha^*\in\mathcal A$ and $v>0$ means that for any $n\in\mathbb N$ and any $v\in M_n(\mathcal A)$, 