Correlation between the weak solutions of a differential equation and implied differential equations

Yes,this is very similar to a previous question I asked. That was about normal solutions and not weak solutions.

We define the operator known as the implied derivative denoted as $I(f)(x)(g)$ to be:

$$I(f)(x)(g) := g(x) \left(\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h} \right) + (1-g(x)) \left(\lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h} \right)$$

Where $g(x)$ is an arbitrary characteristic/indicator function.

I wish to prove whether or not the following conjecture is true. I have no idea how to go about doing that.

Is a function a weak solution to an ordinary differential equation if and only if it is a continuous solution to the corresponding implied differential equation?

By corresponding equations, I just mean that they are corresponding if they are the same except with all of the derivative operators replaced with the implied derivative operator.