Let $u:[0,1]\to\mathbb{R}^n$ be a bounded Borel function.
It is well-known that if, for any compact interval $I\subseteq [0,1]$,
$$ \int\_I|u-u\_I|^2\le C|I|^{1+\alpha} $$
for some $C,\alpha>0$ (here $u\_I:=\frac{1}{|I|}\int\_I u$), then $u$ is in fact $\frac{\alpha}{2}$-Holder continuous: this was first proved by Campanato in 1963.


**Q:** Is it true that if
$$ \int\_I|u-u\_I|^2\le |I|\omega(|I|) $$
for any $I$ then $u$ is continuous? Here $\omega$ denotes an arbitrary modulus of continuity. If this is false in general, can one characterize the $\omega$'s for which this is true?



Does infinitesimal variance imply continuity?
Since $u$ is assumed bounded, your condition is equivalent to
$$
\frac{1}{|I|} \int\_I |u-u\_I|\, dx =o(1)
$$
as $|I|\to 0$, uniformly in $I$, and this is the condition that defines VMO.


So you are asking if functions in $VMO\cap L^{\infty}$ are continuous, and this is known to be false. [This classical paper by Sarason](http://www.ams.org/journals/tran/1975-207-00/S0002-9947-1975-0377518-3/) introduced VMO; Theorem 1(iv) there answers your question, modulo facts about the Hilbert transform. Here's a more direct [reference.](https://books.google.com/books?id=NJkRBwAAQBAJ&pg=PA157&lpg=PA157&dq=banach%20algebra%20QC&source=bl&ots=tTv00lfJv5&sig=wJ9f-KQZmdc2bUynEZi7yDQUKJE&hl=en&sa=X&ved=0ahUKEwj4yPWWzsHLAhUJ32MKHQX5DC4Q6AEIHjAB#v=onepage&q=banach%20algebra%20QC&f=false)


