When logic and calculus collide

[Lil' Lavery]

A few days ago we were given this problem as a warm-up in calculus,
"Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same velocity."
Well, if you consider the position equation f(t)=g(t)-h(t) (g being the position of one runner and h the other). Then the velocity equation would be f'(t)=g'(t)-h'(t), so if the velocities are the same, f'(t)=0. When t=0 and when the race end (another value t), and possibly other ponts, f(t)=0. And because the derivative of a constant is 0, then f'(t) would also be 0. Another way to look at it would be that both runners ran the same distance in the same amount of time, and therefore had the same average velocity, therefore g'(t)=h'(t), and once again f'(t)=0.
But, logically, does it make sense? Other than at the start (which we assume would have a velocity of 0), would the two runners be running at the same speed, at the same time? They would both have to run at the average velocity at one point or another, but would that (or any other speed) be at the exact same moment?