Thursday, April 8, 2010  The Intermediate Under-Valued Theorem
Bruce Peterson - Middlebury College  “Well, duh.”  A familiar
student reaction to the Intermediate Value Theorem. After all,
if a function is “continuous,” it can’t jump from place to place
without stopping in between. Or,  “Real functions are like x2or
sin(x). Those step thingies don’t really matter.”  This
cherished theorem usually falls flat in beginning calculus
because, I would argue, students see it as so obvious as not to
merit discussion. And it’s not their fault; the theorem that
justifies the word “continuous” strikes most students as
unimportant because they rarely see it do anything other than
confirm their long-held intuition about what “continuous” ought
to mean. If a continuous function is positive somewhere and
negative somewhere else then, sure, it has a root in between.
But this familiar “application” is of course just a
restatement—or a special case—of the original result.  So what
kinds of applications are there? For starters,