  We consider the behavior of the price of anarchy and equilibrium flows in
nonatomic multi-commodity routing games as a function of the traffic demand. We
analyze their smoothness with a special attention to specific values of the
demand at which the support of the Wardrop equilibrium exhibits a phase
transition with an abrupt change in the set of optimal routes. Typically, when
such a phase transition occurs, the price of anarchy function has a breakpoint,
\ie is not differentiable. We prove that, if the demand varies proportionally
across all commodities, then, at a breakpoint, the largest left or right
derivatives of the price of anarchy and of the social cost at equilibrium, are
associated with the smaller equilibrium support. This proves -- under the
assumption of proportional demand -- a conjecture of O'Hare et al. (2016), who
observed this behavior in simulations. We also provide counterexamples showing
that this monotonicity of the one-sided derivatives may fail when the demand
does not vary proportionally, even if it moves along a straight line not
passing through the origin.
