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196
CHAPTER 2. CURVES
(c)
Show
that if
x
0
is a
triadic rational
(that is, it has the form
x
0
=
p
3
j
for some
j
) then
f
k
+1
(
x
0
) =
f
k
(
x
0
) for
k
suFciently
large, and hence this is the value
f
(
x
0
). In particular,
show
that
f
has a local extremum at each triadic rational. (
Hint:
x
0
is a local extremum for all
f
k
once
k
is suFciently large;
furthermore, once this happens, the sign of the slope on either
side does not change, and its absolute value is increasing with
k
.
)
This shows that
f
has in±nitely many local extrema—in fact,
between any two points of [0
,
1] there is a local maximum (and a
local minimum); in other words, the curve has in±nitely many
“corners”. It can be shown (see (
Calculus Deconstructed
,
§
4.11)) that
the function
f
, while it is continuous on [0
,
1], is not di²erentiable at
any point of the interval. In Exercise
5
in
§
2.5
, we will also see that
this curve has in±nite “length”.
2.5
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 Fall '08
 ALL
 Calculus

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