Can sets of points have dimensions that differ from point to point?

Question

I have thought about something interesting, in regards to dimension and Euclidean space $\mathbb{R}^n$. To take a simple case, consider $\mathbb{R}^3$, and take the set which is the union of the $xy$-plane and the $z$-axis. At any point in the $xy$-plane other than the origin, the dimension is $2$, and at any point in the $z$-axis other than the origin, the dimension is $1$, and at the origin, the dimension is $2$, because that is the maximum of $1$ and $2$. So, basically, I am talking about a function that takes a subset $S$ of $\mathbb{R}^n$, and assigns to each point $p$ of $S$ a number that is the dimension. My real question is, has anyone defined a similar concept to what I am suggesting, and developed the theory quite a bit? I would love to see a text that talks about this or a very similar concept.