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We expand upon the basic oscillation theory for general boundary problems of the form $$-y''+q(t)y=\lambda r(t)y, \quad t \in I = [a,b] $$ where q and r are real-valued piecewise continuous functions and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. The non-definite case is characterized by the indefiniteness of each of the quadratic forms $$ B+\int_a^b (|y'|^2 +q|y|^2)\quad \text{and}\quad \int_a^b r|y|^2, $$ over a suitable space where B is a boundary term. In 1918 Richardson proved that, in the case of the Dirichlet problem, if r(t) changes its sign exactly once and the boundary problem is non-definite then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.