  We address a classical open question by H.Brezis and R.Ignat concerning the
characterization of constant functions through double integrals that involve
difference quotients.
  Our first result is a counterexample to the question in its full generality.
This counterexample requires the construction of a function whose difference
quotients avoid a sequence of intervals with endpoints that diverge to
infinity. Our second result is a positive answer to the question when
restricted either to functions that are bounded and approximately
differentiable almost everywhere, or to functions with bounded variation.
  We also present some related open problems that are motivated by our positive
and negative results.
