 Oscillation of the Buchstab function

The Buchstab function takes values both above and below $e^{-\gamma }$ on every interval of length one in $[1,\infty )$. The equation $\frac{d}{du}(u(w(u)-e^{-\gamma }))=w(u-1)-e^{-\gamma }$ propagates any fixed sign forwards; the rapid decay to zero then forces an impossible identically zero tail.