A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema

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In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function 𝑓 which is the uniform limit of a sequence of sawtooth functions 𝑓ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let 𝑋 = [0, 1] x [0, 1] and 𝐹(𝑋) be the Hausdorff metric space determined by 𝑋. We define contraction maps 𝑤₁ , 𝑤₂ , 𝑤₃ on 𝑋. These maps define a contraction map 𝑤 on 𝐹(𝑋) via 𝑤(𝐴) = 𝑤₁(𝐴) ⋃ 𝑤₂(𝐴) ⋃ 𝑤₃(𝐴). The iteration under 𝑤 of the diagonal in 𝑋 defines a sequence of graphs of continuous functions ... continued below

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iv, 81 leaves: ill.

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Huggins, Mark C. (Mark Christopher) December 1993.

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  • Huggins, Mark C. (Mark Christopher)

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In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function 𝑓 which is the uniform limit of a sequence of sawtooth functions 𝑓ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let 𝑋 = [0, 1] x [0, 1] and 𝐹(𝑋) be the Hausdorff metric space determined by 𝑋. We define contraction maps 𝑤₁ , 𝑤₂ , 𝑤₃ on 𝑋. These maps define a contraction map 𝑤 on 𝐹(𝑋) via 𝑤(𝐴) = 𝑤₁(𝐴) ⋃ 𝑤₂(𝐴) ⋃ 𝑤₃(𝐴). The iteration under 𝑤 of the diagonal in 𝑋 defines a sequence of graphs of continuous functions 𝑓ₙ. Since 𝑤 is a contraction map in the compact metric space 𝐹(𝑋), 𝑤 has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function 𝑓. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in 𝐶[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set of proper local extrema is residual in 𝐶[0,1]. In the fourth and last chapter we actually construct our function and prove it is continuous, nowhere-differentiable and has a dense set of proper local extrema. Lastly we iterate the set {(0,0), (1,1)} under 𝑤 and plot its points. Any terms not defined in Chapters 2 through 4 may be found in [2,4]. The same applies to the basic properties of metric spaces which have not been explicitly stated. Throughout, we will let 𝒩 and 𝕽 denote the natural numbers and the real numbers, respectively.

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iv, 81 leaves: ill.

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  • December 1993

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  • March 9, 2015, 8:15 a.m.

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  • June 20, 2017, 12:20 p.m.

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Huggins, Mark C. (Mark Christopher). A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema, thesis, December 1993; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc500353/: accessed November 17, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .