The following Theorem is from the book Theory and Applications of Fractional Differential Equations by Kilbas, Srivastava and Trujillo. It's Theorem 3.10 from page 163.
Theorem. Let $ \alpha \in \mathbb{R} $, $ \alpha > 0 $ with associated $ n = - \lfloor- \alpha \rfloor $ and $ b_{k} \in \mathbb{R} $ for $ k=1,\dots,n $. Further, let $ G $ be an open set in $ \mathbb{R} $ and $ f:(a,b] \times G \to \mathbb{R} $ s.t. $ f(\cdot,y) \in C_{n - \alpha}(a,b] $ for all $ y \in G $. If $ g \in C_{n - \alpha}(a,b] $, then $ g $ satisfies the Cauchy type problem
\begin{align} (D^{\alpha}_{a+} g)(x) &= f(x,g(x)), &x \in (a,b],\label{eq:EquivCTPDifEq} \\ (D^{\alpha - k}_{a+} g)(a+) &= b_{k}, &k=1,\dots,n,\label{eq:EquivCTPInitVal} \end{align}
if, and only if $ g $ satisfies the nonlinear Volterra integral equation of the second kind
\begin{equation} g(x) = \sum_{j=1}^{n} \frac{b_{j}}{\Gamma(\alpha - j + 1)} (x-a)^{\alpha - j} + \frac{1}{\Gamma(\alpha)} \int_{a}^{x} \frac{f(t,g(t))}{(x-t)^{1-\alpha}}\, d t, \quad x \in (a,b]. \end{equation}
Question. The proof of necessity starts by arguing, if $ g $ satisfies the Cauchy type problem, $ (D^{\alpha}_{a+} g)(x) = f(x,g(x)) $ means that the fractional derivative, $ (D^{\alpha}_{a+} g)(x) $, exists and is in $ C_{n - \alpha}(a,b] $, since $ f(\cdot,y) \in C_{n - \alpha}(a,b] $. Yet, in my opinion, $ f(\cdot,y) \in C_{n - \alpha}(a,b] $ for all $ y \in G $ and $ g \in C_{n - \alpha}(a,b] $ is not enough to ensure $ f(x,g(x)) \in C_{n - \alpha}(a,b] $ for $ x \in (a,b] $. Am I missing something obvious? I think more assumptions on $ f $ are needed to justify that argument.
Definitions
$ (D^{\alpha}_{a+} g)(x) $ is the Riemann-Liouville fractional derivative.
For $ -\infty < a < b < \infty $ and $ \gamma \in \mathbb{C} \ (0 \le \Re(\gamma) < 1)$ the space $ C_{\gamma}(a,b] $ stands for weighted functions $ f $ given on $ (a,b] $ such that $ (x-a)^{\gamma}f(x) \in C[a,b] $, i.e. $ \lim_{x \to a+} (x-a)^{\gamma}f(x) $ exists.
