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Topic: I'm confused
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Joshua Zucker

Posts: 710
Registered: 12/4/04
I'm confused
Posted: Apr 27, 1998 9:29 PM
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I am also confused about this issue of open vs. closed intervals, so I
figured I'd try to answer this fine question to see if I could
straighten out my own thinking.

One way to look at the property of being an increasing function is
that if you go from a up to b, the function value also increases.
That is, a function is increasing on an interval if for any a < b in
the interval, f(a) < f(b) (or maybe = if not strictly increasing...)
Now, that's true on a closed interval including the max and min of the
function; that is, for sin, on [-pi/2, pi/2] it has this property.

On the other hand, when you think of increasing not as a property of
two DIFFERENT points in the interval, but rather as a property
happening AT a single point, then certainly sin is not increasing at
pi/2, its derivative is 0 there and it is a maximum point. You can't
increase past a maximum! So in this interpretation, where we FIRST
classify each individual point according to what's going on locally
there and THEN group them into intervals based on that, the set of
increasing points would be a union of open intervals like (-pi/2, pi/2).

Did I make any sense?
--Joshua Zucker

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