The paper "Wituła, R.; Hetmaniok, E.; Słota, D., Mean-value theorem for one-sided differentiable functions, Fasc. Math. 48, 143-154 (2012). ZBL1259.26003." discusses some generalizations of the Mean Value Theorem (MVT) when one considers the setting where a function has one-sided derivatives but may not be differentiable.
They state the following Theorem:
If $f:[a, b] \rightarrow \mathbb{R}$ is both-sided differentiable in $(a,b)$, right-continuous at $a$, left-continuous at $b$, then there exists $c \in (a, b), p, q \geq 0, p + q = 1$ such that $\frac{f(b) - > f(a)}{b-a} = p f'(c+) + qf'(c-)$ where $f'(c+)$ denotes the right derivative and $f'(c-)$ the left derivative.
And then remark that based on this Theorem the following inequality holds:
There exists $c \in (a, b)$ sucht hat
$\min \{ f'(c-), f'(c+) \} \leq \frac{f(b)-f(a)}{b-a} \leq \max \{ f'(c-), f'(c+) \}$
I am now wondering whether something similar can be stated in the multivariate case where $f: S \subset \mathbb{R}^n \rightarrow \mathbb{R}$ only has one-sided partial derivatives.
The MVT in multivariate case for differentiable $f: S \subset \mathbb{R}^n \rightarrow \mathbb{R}$ is stated as:
For two points $a, b \in S$, let $L_{a, b}$ denote the line segment that connects them. If $L_{a,b} \subseteq S$ there exists $c \in L_{a,b}$ such that
$$f(b)-f(a) = (b-a)^T \nabla f(c)$$
If we now suppose that $f$ is not differentiable but only has one-sided partial derivatives, is there an inequality similar to the case where $f$ is univariate?