  We consider the energy norm arising from elliptic problems with discontinuous
piecewise constant diffusion. We prove that under the quasi-monotonicity
property on the diffusion coefficient, the best approximation error with
continuous piecewise polynomials is equivalent to the $\ell_2$-sum of best
errors on elements, in the spirit of A. Veeser for the $H^1$-seminorm. If the
quasi-monotonicity is violated, counterexamples show that a robust localization
does not hold in general, neither on elements, nor on pairs of adjacent
elements, nor on stars of elements sharing a common vertex.
