Let $I = [0,1]$, $\phi, \psi: I \rightarrow \mathbb{R}$ functions. Does $\phi$ exist such that any $\psi$ adhering to the following inequality is non-measurable: $$\sup\_{x \in I} |\phi(x) - \psi(x)| \le 1?$$ Assuming so, is it possible to replace 1 with $C$ representing a non-negative constant (and are there any bounds on what value $C$ can take)?

In particular, taking $\psi = \phi$, this criterion should force $\phi$ to be non-measurable itself. Besides risking this observation, I am at a loss for establishing such a $\phi$. I was trying to work with characteristic functions (i.e.,for non-measurable sets) since these can afford surprisingly-useful examples; nothing crystallized, though, so perhaps I went in a bad direction. I hope that someone visiting the site could assist me here. The second question is more of a personal question/ "philosophical" question for edification. Thanks in advance for any assistance you can give (it seems really curious that measure theory could yield a function with such a property)!