Crossing of smooth schedules

Suppose demand $D(p)$ is continuous and strictly decreasing and supply $S(p)$ continuous and strictly increasing on an interval, with $D(p_0) > S(p_0)$ and $D(p_1) < S(p_1)$ at its ends. Show that $V(p) = \min\{D(p), S(p)\}$ has a unique maximiser, and that it is the unique $\hat p$ with $D(\hat p) = S(\hat p)$. Explain in one paragraph why the discrete auction in 1.3 needs tie-break rules that this continuous version does not.