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188 ChapterS SequencesandSeriesofFunctions

integerN such that fut Q, < e l18M) forall n > N.If x e [0, 1] andn > N,then

lP"(x) - f (x) l :

Hence, the sequence {P,} converges uniformly to / on [0, 1]. This completes theproof. r

The above proof, although not tenibly difficult, is not all that enlightening. Itis not easy to actually find the polynomials that approximate the function, and theconvergence of the polynomials is difficult to visualize. Neveftheless, a sequenceof polynomials that converges uniformly to / on [a, b] does exist.

The second result is even more surprising than the first. It states that there exists acontinuous, nowhere differentiable function. An example of such a function was firstpublished by Weierstrass in 1872, and it created quite a stir among mathematicians.It had been taken for granted that continuous functions were differentiable at mostpoints; think about the type of graph you normally draw to represent a continuousfunction. After rigorous definitions for continuity offunctions and convergence ofseries were given, it was possible to see where these definitions led. It is imperativethat a simple mental picture of a continuous function be set aside; a continuousfunction is a function that satisfies the def,nition of continuity. Results contrary tointuition sometimes appear. When this occurs, either the definition has been poorlyformulated (and thus needs to be altered) or intuition needs to be expanded to includenew possibilities. In this case, since the definitions of continuity and convergenceare well established, it is the intuition that must adapt.

A construction of a continuous, nowhere differentiable function is given below.(This example is different than the one published by Weierstrass.) This constructionuses a common device for creating continuous functions. Any function that is theuniform limit of a series of continuous functions is itself continuous.

THEOREM 8.12 There exists a continuous function that is not differentiable atany point.

Proof. Define a function g:lR. -+ IR by letting g@) : lxl for -I < x < 1 andS@ -+ 2) : g(x) for all other values of x. (The graph of g can be found in Figure8.4.) By definition, the function g is continuous on IR, 0 = g(x) < I for all x, and

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