Want to find a subset $X$ of $\mathbb{R}^3$ such that there exists a cts function $f$ on the set such that the image $f(X) = \{0,1\}$, that is takes both values 0 and 1.
This is part of a larger question (IVT in higher dimensions) but just wanted to help understand this portion. Should be the case that some subsets of $\mathbb{R}^3$ have such a function while others don't.
For a subset that doesn't, I came up with the set that consists only of the zero vector, ${(0,0,0)}$.
I might be overthinking it but struggling to come up with one since the function that exists needs to be continuous (otherwise I'd just do a piece wise function).
Appreciate any tips
Hint: A crucial hypothesis in IVT is connectedness. Can you think of such a function on a disconnected domain?