I recently came across a generalized Implicit Function Theorem (Jittorntrum (1978) and Kumagai (1980)) that relaxes the requirements of differentiability and non-singularity (essentially just requires the function to be continuous and locally injective/one-to-one). The following question came to mind: is there an an analogue of implicit differentiation for one-sided derivatives?
For concreteness, consider $F(x,y) = 0$ where $F:\mathbb{R}^2\to\mathbb{R}$ is a continuous function. Suppose that for some ($x_0,y_0)\in\mathbb{R}^2$, there exist $\varepsilon_x,\varepsilon_y>0$ s.t. for every fixed $y\in(y_0-\varepsilon_y,y_0+\varepsilon_y)$, $F(\cdot,y):(x_0-\varepsilon_x,x_0+\varepsilon_x)\to\mathbb{R}$ is locally one-to-one. (This supposition is just so that Theorem 2.1 from Jittorntrum (1978) holds).
Now, let $\partial_x^- F(x,y) = \lim_{h\nearrow 0} \frac{F(x+h,y)-F(x,y)}{h} $ denote the left-hand partial derivative w.r.t. $x$ and define $\partial_y^- F(x,y)$ similarly. Then, is the following true
$$ \partial_x^-F(x,y)\cdot\underset{=1}{\underbrace{\partial_x^- x}} + \partial_y^- F(x,y) \cdot \partial_x^- y = 0 $$
for all $(x,y) \in (x_0-\varepsilon_x,x_0+\varepsilon_x) \times (y_0-\varepsilon_y,y_0+\varepsilon_y) $?
Thanks so much for reading! Any insights for why the above is (or is not) true would be very much appreciated!