### Continuity Theorems

#### Same Time, Same Place

##### A Story In Which You Meet Yourself
Early one morning you decide to climb an inviting knoll not too far in the distance. When you reach its base you find a well-marked trail meandering up its side. At exactly 9:00 you start up the trail. After a leisurely stroll including numerous stops, snacks, and a long lunch break, you reach the top at 5:00 in the afternoon.

It seems best to spend the night, which you do. The next morning you are feeling a bit lazy and it is once again 9:00 before you start back down. This turns out to be a mistake because by 10:00 clouds are gathering and by 11:00 you are walking in a light but annoying rain. Given the weather, you decide not to take any breaks but rather to keep slogging down the knoll. Unfortunately, the going is slippery and slow, and it is coincidentally exactly 5:00 when a water-logged and thoroughly miserable version of your former self reaches the bottom of the knoll.

The next day you get a thorough talking-to about weather and preparedness, but your guide also brings up what seems to be a coincidence. "You know", says your guide, "from what you have told me there is a place on the trail which you passed at exactly the same time on your way up as on your way down."

### Why is THAT true?

There are a number of ways to see why your guide's claim must be true. One clever way is to picture yourself at the bottom of the trail and your water-logged twin at the top of the trail. Both versions of yourself take off at 9:00. By 5:00 they have switched places. Since neither left the tail, they must have met along the way.

Another way to substantiate the guide's claim is to turn your trip into a mathematical model. For this purpose we consider the points on the trail as laid out on the y-axis, with 0 indicating the bottom of the trail and 1 corresponding to the top. The point .5 corresponds to the point half way up the trail, the point .75 corresponds to the point three-quarters of the way up the trail, and so on.

 Do I Understand the Graph?
The x-axis represents time, running from 9:00 AM to 5:00 PM. This allows us to graph your trip up the knoll (blue) and your trip down the knoll (green) on one graph. Your guide's conclusion is that the graphs must cross at least once. In the picture, they actually cross 3 times (shown by red circles.)

In making conclusions such as "the graphs must cross at least once" your guide is using the fact that the functions are continuous. This assumption of continuity is hidden in surprisingly many "obvious" conclusions.

In this section we study two theorems about continuous functions. The first is the one which we use to conclude that graphs such as the ones above must cross. This is the Intermediate Value Theorem (IVT). It can be used (with the aid of a microchip, usually) to find the zeros of a continuous function and hence to compute the decimal expansion of numbers like sqrt(2). The second theorem is the Extreme Value Theorem (EVT) which says that a continuous function has a largest and smallest value on a closed interval.

Your objective in this section is to understand

• The statements and content of the IVT and EVT.

• The reason that continuity is necessary for these theorems to be valid. (Counterexamples showing what goes wrong if the functions are NOT continuous are very useful)

• How to apply the IVT and EVT.
We will not discuss proofs of any of these theorems.

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