I'm studying multivariable calculus, and I was wondering:
Is there a way to characterize all sets in $\mathbb{R}^2$ that are continuously parametrizable? What I mean with a continuously parametrizable set $A$ is that there exists a continuous function $f:[0, 1] \rightarrow \mathbb{R}^2$ onto $A$.
I know that curves are usually defined as continuous functions from intervals, so I'm looking for something different than what was asked here. I also found this question related, but it uses manifolds, so I was wondering if there is something easier to understand.
My thoughts on this question:
I know that the continuous image of a connected and compact set is connected and compact. So the sets I'm looking for must be connected and compact. Furthermore, since the graph of continuous functions has measure zero (see here), my intuition is that (although I'm not looking for sets that are graphs of functions), maybe the only sets that satisfy the desired property have measure zero. Can anyone help me to know if there's a result about this question, or provide me with help, please?