WeierstrassFunction

WeierstrassFunction - THE WEIERSTRASS PATHOLOGICAL FUNCTION...

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THE WEIERSTRASS PATHOLOGICAL FUNCTION Until Weierstrass published his shocking paper in 1872, most of the mathematical world (including luminaries like Gauss) believed that a continuous function could only fail to be differentiable at some collection of isolated points. In fact, it turns out that “most” continuous functions are non-differentiable at all points. (To understand what this state- ment could mean, you should take courses in topology and measure theory.) However, Weierstrass was not, in fact, the first to construct such a pathological function. He was preceded by Bolzano (in 1830), Cell’erier (also 1830), and Riemann (1862). None of the others published their work (indeed, their examples were not discovered in their notes until after their deaths). All known examples of non-differentiable continuous functions are constructed in a similar fashion to the following example – they are limits of functions that oscillate more and more on small scales, but with higher-frequency oscillations being damped quickly. The example we give here is a faithful reproduction of Weierstrass’s original 1872 proof. It is somewhat more complicated than the example given as Theorem 7.18 in Rudin, but is superior in at least one important way, as explained in Remark 2. Theorem 1. Let 0 < a < 1 , and choose a positive odd integer b large enough that π ab - 1 < 2 3 (i.e. ab > 1 + 3 π 2 ). For example, take a = 1 2 and b = 11 . Define the function W : R R by W ( x ) = X n =0 a n cos( b n πx ) . Then W is uniformly continuous on R , but is differentiable at no point. Remark 2 . Notice that the partial sums W N ( x ) = N n =0 a n cos( b n πx ) are all C functions. As the following proof shows, these partial sums converge uniformly to W , and so we have an example here of a sequence of C functions that converge uniformly to a nowhere- differentiable function. This is the most dramatic demonstration that differentiability is not preserved under uniform convergence! The example of Theorem 7.18 in Rudin, while
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WeierstrassFunction - THE WEIERSTRASS PATHOLOGICAL FUNCTION...

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