Alice and Bob both live in point A. Both work in point B.
Every morning, the time it takes to get from A to B, starts at value X, goes up (with rush hour) to Y and then back to X (for our discussion, let's not define how it gets from X to Y and back but it is a continuous function).
Alice and Bob want to see if they can leave at different times in the morning and arrive at the same time. Assume that both drive the highest speed they can give the traffic, and no more than the speed limit.
So think about it this way: There is a function f that maps the departure time d (in seconds since midnight) to how long the trip would be in seconds (call it l for length). Is it possible for:
Can you prove if they can, or cannot? If they can, when?
For example, if the function is structured so that each second later that you depart, the trip gets shorter by a second, it's possible (essentially the function is behaving like $-d+b$ in that specific section). But can a function like this exist in the situation of traffic patterns?