I am interested in a mean value theorem (MVT) for vector valued functions which are not $C^1$ maps. I motivate the problem in the case of a $C^1$ map. Then I try to derive an MVT for the case of Lipschitz functions. My questions refer to the Lipschitz case and how we can generalize it further.
Motivation of the problem: the case where $f$ is a $C^1$ map
Let $U$ be an open subset of $\mathbb{R}^p$ ($1 < p < \infty$) and let $\boldsymbol{x}_0, \boldsymbol{h} \in U$. Denote by $\boldsymbol{f} : U \rightarrow \mathbb{R}p$ a $C^1$ map; the Jacobian matrix of $\boldsymbol{f}$ is indicated by $\boldsymbol{f}'(\boldsymbol{x})$. Suppose that the line segment $\boldsymbol{x}_0 + t \boldsymbol{h}$, $0 \leq t \leq 1$, is contained in $U$. By the mean value theorem (MVT) for vector-valued functions (cf. Thm. 4.2, Lang, 1993, p. 341), we have
\begin{equation} \boldsymbol{f}(\boldsymbol{x}_0 + \boldsymbol{h}) - \boldsymbol{f}(\boldsymbol{x}_0) = \Big[ \int_0^1 \boldsymbol{f}'( \boldsymbol{x}_0 + u \boldsymbol{h}) \mathrm{d} u \Big] \boldsymbol{h}. \tag{1} \end{equation}
Serge Lang gives the following proof. Consider the auxiliary function $\boldsymbol{v}:[0,1] \rightarrow U \subset \mathbb{R}^p$ given by $\boldsymbol{v}(t) = \boldsymbol{f}(\boldsymbol{x}_0 + t \boldsymbol{h})$. Then $\boldsymbol{v}'(t) = \partial \boldsymbol{v}(t) / \partial t = \boldsymbol{f}'(\boldsymbol{x}_0 + t \boldsymbol{h}) \boldsymbol{h}$. By the fundamental theorem of calculus (FTC), we have
\begin{equation*} \boldsymbol{v}(1) - \boldsymbol{v}(0) = \int_0^1 \boldsymbol{v}'(t) \mathrm{d} t. \end{equation*}
As $\boldsymbol{v}(0)=\boldsymbol{f}(\boldsymbol{x}_0)$ and $\boldsymbol{v}(1)=\boldsymbol{f}(\boldsymbol{x}_0 + \boldsymbol{h})$, equation (1) obtains [taking into account another lemma that allows us to pull $\boldsymbol{h}$ out of the integral in (1)].
[Remark: The Assertion of Thm. 4.2. in Lang (1993) actually refers to $C^1$ Banach-valued maps on $U \subset E$, where $E$ denotes a Banach space. I stick with Euclidean space as I want to work with coordinate-wise representations in what follows.]
Problem: Is there an MVT for functions under weaker than the $C^1$ hypothesis?
I wonder under what alternative hypothesis on $\boldsymbol{f}$ we can maintain an MVT (having dropped the $C^1$ assumption). In view of the above proof, I recognize that the ''essential'' part is the FTC. First of all, I restrict attention to Lipschitz functions $\boldsymbol{g}: U \rightarrow \mathbb{R}^p$ with a real-valued $K$ such that for all $\boldsymbol{x}, \boldsymbol{y} \in U$, we have $\Vert \boldsymbol{g}(\boldsymbol{x}) - \boldsymbol{g}(\boldsymbol{y}) \Vert \leq K \Vert \boldsymbol{x} - \boldsymbol{y} \Vert $.
Now, my reasoning in order to obtain an MVT for $\boldsymbol{g} \in \mathrm{Lip}$ is the following:
(i) By Rademacher's thm., $\boldsymbol{g} \in \mathrm{Lip}$ is totally differentiable at $\boldsymbol{x}_0$ for almost every (a.e.) $\boldsymbol{x}_0 \in U$, and $\Vert \boldsymbol{g}(\boldsymbol{x}_0)' \Vert \leq K$ for all $\boldsymbol{x} \in U \setminus Z$, where $Z$ denotes the set (with Lebesgue measure zero) of points in $U$ where $\boldsymbol{g}$ is not differentiable. As total differentiability implies directional differentiability, the directional derivative of $\boldsymbol{g}(\boldsymbol{x}_0)$ in the direction $\boldsymbol{h} \in U$ is given by $\boldsymbol{D}_h \boldsymbol{g}(\boldsymbol{x}_0) = \boldsymbol{g}'(\boldsymbol{x}_0)\cdot \boldsymbol{h}$ for a.e. $\boldsymbol{x}_0 \in U$.
(ii) Let the auxiliary function $\boldsymbol{\varphi}: [0,1] \rightarrow U \subset \mathbb{R}^p$ be defined as $\boldsymbol{\varphi}(t) := \boldsymbol{x}_0 + t \boldsymbol{h}$, where $\boldsymbol{x}_0, \boldsymbol{h} \in U$. Consider the function $\boldsymbol{v}:[0,1] \rightarrow \mathbb{R}^p$ given by the composition $\boldsymbol{v} := \boldsymbol{g} \circ \boldsymbol{\varphi}$ (which is Lipschitz). Note that $\partial \boldsymbol{v}(t) / \partial t = \boldsymbol{v}(t)' = \boldsymbol{g}'(\boldsymbol{x}_0 + t \boldsymbol{h}) \boldsymbol{h} \equiv \boldsymbol{D}_h \boldsymbol{g}(\boldsymbol{u}) \vert_{\boldsymbol{u} = \boldsymbol{\varphi}(t)}$ [i.e. the directional derivative of $\boldsymbol{g}$ in the direction $\boldsymbol{h} \in U$ evaluated at $\boldsymbol{\varphi}(t)$] except on the null set $Z$.
(iii) As $\boldsymbol{v}$ maps $[0,1]$ to $\mathbb{R}^p$, we shall consider its $i=1,\ldots,p$ component functions $v_i:[0,1] \rightarrow \mathbb{R}$ separately; the same applies to $\boldsymbol{v}'(t)$. By the FTC for Lebesgue integration, we have
\begin{equation*} v_i(1) - v_i(0) = \int_{[0,1]} v_i' \lambda \qquad \text{for all} \quad i = 1, \ldots, p, \tag{A} \end{equation*} where $\lambda$ denotes the Lebesgue measure.(iv) Next, we define integration over the vector-valued function $\boldsymbol{v}'(t)$ as component-wise integration, \begin{equation*} \int_{[0,1]} \boldsymbol{v}' \mathrm{} \lambda := \Big( \int_{[0,1]} v_1' \mathrm{d} \lambda, \ldots, \int_{[0,1]} v_p' \mathrm{d} \lambda \Big). \tag{B} \end{equation*}
(v) In view of (A), (B), and the definitions of $\boldsymbol{v}$ and $\boldsymbol{v}'$, we have for $\boldsymbol{x}_0, \boldsymbol{h} \in U$, \begin{equation*} \boldsymbol{g}( \boldsymbol{x}_0 + \boldsymbol{h}) - \boldsymbol{g}( \boldsymbol{x}_0) = \int_{[0,1]} \boldsymbol{g}'(\boldsymbol{x}_0 + t \boldsymbol{h}) \lambda(t) \boldsymbol{h}. \tag{C} \end{equation*} Hence, (C) establishes an MVT for Lipschitz functions.
Questions
Q1: Is my reasoning (sketch of proof) correct? Is there an approach that is simpler / more straightforward?
Q2: I wonder whether we can formulate a similar MVT for absolutely continuous functions; however, I struggle with the definition of absolute continuity of vector-valued functions. Can you give me a hint how to attack this problem?