A counterexample to a theorem about the convexity of Powell-Sabin elements

The function F has three semi-infinite lines of tangent discontinuity: l1 = {(x, 0) : x ≥ 0}, l2 = {(x,−x) : x ≤ 0}, and l3 = {(x, x) : x ≤ 0} which meet at the origin. These lines, when restricted to the triangle △, divide △ into three quadrilaterals. If we further divide these quadrilaterals by the three line segments [0,x1], [0,x2], [0,x3] joining the origin to the three vertices of △ we obtain a Powell-Sabin split of △ into six triangles, shown in Figure 1. It was shown by Powell and Sabin (1977) that there is a unique C function f : △ → IR whose restriction to each of the six triangles is a quadratic polynomial and which satisfies