Volume delay function

In a Traffic Assignment Problem, a volume delay function is a function that gives the cost for a user to use a link of a network, taking into account the congestion of the link when that user arrives.

Given a flow on a link equal to $x$, the volume delay function is:

and it's equal to the cost of $x$ users to use the link.

Each user $x_i$ that enters the link experiences a new cost $s(x_i)$. The cumulative cost experienced by all users would be:

If we assume there are enough users that we can consider $x$ to be a continuous variable, $x\in \mathbb{R}$, then, the total cumulative cost, with a flow $x$ in the link, would be: