On the continuity of functions in $W^{1}_{p}$ which are monotonic in one direction

Abstract:

It was previously shown that, for $n = 2$, if f is such that its distribution derivatives are measures, and f is monotonically nondecreasing in one variable for almost all values of the other variable then f is equivalent to a continuous function. This is now shown to be false for $n > 2$. It is true for $f \in W_p^1,p > n - 1$ and may be false for $f \in W_{n - 1}^1$.