Despite being common to have a continuous function that is not a differentiable function, in engineering and applied mathematics, it's quite common to have non-differentiable functions that are related to analytic functions, like semianalytic functions. Such type of functions satisfies stronger properties than the continuous functions, like the Lojasiewicz inequality.
Thinking on applications, for a semianalytic, see Definition 2.1. (or subanalytic, see Definition 3.1.), Lipschitz continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, my question is whether a "weak notion of continuity of the derivative" (given ahead in (*)) holds for such types of functions. Expressly, defining the set of differentiable points of $f$ (which has full measure in $\mathbb{R}^{n}$) as $\Delta_{f}$, I want to know if, for all $x^{*} \in \mathbb{R}^{n}$, and all $ \epsilon > 0$, there exist $ \delta > 0 $ such that for all $y^{*} \in B(x^{*},\delta)$ it holds $$\begin{align} \lim_{\Delta \to 0} \ \inf \{ \|\nabla f(x) - \nabla f(y) \| : \\ x \in \Delta_{f} \cap B(x^{*};\Delta)\setminus\{x^{*}\} \text{ and } y \in \Delta_{f} \cap B(y^{*};\Delta)\setminus\{y^{*}\} \} <\epsilon \tag{*} \end{align}$$
Note that, when the function is locally Fréchet differentiable with locally continuous derivative, this is true since it reduces to the continuity of the derivative. Also, It seems to be true for all subanalytic functions.
$$\text{Is the notion of continuity in (*) true for subanalytic or semianalytic functions?}$$