The resistance of a given object is expressed through:
$$R=\rho\frac{l}{A}$$
I'm wondering if there is any quantity like resistance at a specific point. For example, $R$ for a copper wire with l=100um, w=t=50nm can be easily found. But wondering if for example $R(x=50\ \mathrm{\mu m}$ has any meaning or makes sense?
Something like $R(x=50\ \mathrm{\mu m})$ itself isn't meaningful, because resistance is something that "builds up" over the length of the material. Kind of like volume or mass, it depends on how much you have.
You could think like this: if resistance "builds up" over the length of the material, what is it being built up from? Or more precisely, if resistance builds up over length, it should be expressible as the integral of some quantity: $$R = \int_0^l f(x)\ \mathrm{d}x$$ That line of reasoning would lead you to the resistance per unit length, $f(x) = \frac{\partial R}{\partial x}$. This might be the closest thing to what you're asking about: it is meaningful to evaluate this at a certain length along the object, say $f(50\ \mathrm{\mu m})$.
But as long as $R = \rho\frac{l}{A}$ is valid, you'd find that the resistance per unit length is just $\frac{\rho}{A}$. So it's easier to just work with resistivity, which is a property of the material, theoretically independent from all geometric factors like length and area.
