Figure 4.

The reason that the Pareto frontier can never be reached using non-contiguous bins is that swapping parts of them against parts of an intermediate bin can increase $I(Z,X)$ while keeping $H\left(Z\right)$ constant. In this example, the binning function g assigns two separate W-intervals (top panel) to the same bin (bin 2) as seen in the bottom panel. The shaded rectangles have widths $P_i$, heights $p_i$ and areas $P_{i1}=P_i{p}_1$. In the upper panel, the conditional probabilities $p_i$ are monotonically increasing because they are averages of the monotonically increasing curve $p_1\left(w\right)$.