196 CHAPTER 2. CURVES (c) Show that if x0 is a triadic rational (that is, it has the form x0 = p 3 j for some j ) then f k +1 ( x0 ) = f k ( x0 ) for k suFciently large, and hence this is the value f ( x0 ). In particular, show that f has a local extremum at each triadic rational. ( Hint: x0 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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