Rubinstein2005-page90

Rubinstein2005-page90 - 12:18 72 master Sheet number 88...

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October 21, 2005 12:18 master Sheet number 88 Page number 72 72 Lecture Six it must rest for half a day. Then, the maximal distance it can travel in 1 day is 1 km but it can travel this distance in only half a day. b. For (2) to imply (1) we need to assume the turtle cannot “jump” a positive distance in zero time. Contrast this with the case in which the turtle’s speed is 1 km/day but after a day of traveling it can “jump” 1 km. Thus, it needs 1 day to travel 1 km but within 1 day it can travel 2 km. The assumptions that in any positive interval of time the turtle can travel a positive distance and that the turtle cannot “jump” are sufﬁcient for the equivalence of (1) and (2). Let M ( t ) be the maximal distance the turtle can travel in time t . Assume that the function M is strictly increasing and continuous. Then, the statement, “The maximal distance a turtle can travel in t is x ” is equivalent to the statement, “The minimal time it takes a turtle to travel
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