What is the relation between instantaneous speed and instantaneous velocity?

If I drive away from my house at 8:00 in the morning and return at 6:00 PM that same

evening with 50 more miles showing on the car, you know immediately that my average

speed for the day was 5 mph.

But you don't know a thing about how much of that time I was stopped, how much

in motion, or what my speed was at any moment between 8 and 6, because there's

no necessary relationship between instantaneous and average speed.

I guess it's probably true to say that there has to be some instant during any period

of time when the instantaneous speed must be equal to the average speed during

the same period. That sounds like a nice theorem, and its proof ought to be good

for some mathematical recreation, but it doesn't seem too useful.