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University of North Texas

1. Huggins, Mark C. (Mark Christopher). A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema.

Degree: 1993, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc500353/

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.
*Advisors/Committee Members: Bator, Elizabeth M., Lewis, Paul Weldon, Jackson, Steve, 1957-.*

Subjects/Keywords: contraction maps; contractive maps; continuous nowhere differentiable functions; continuous functions; Baire Category Theorem; Functions, Continuous.

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APA (6^{th} Edition):

Huggins, M. C. (. C. (1993). A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc500353/

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Huggins, Mark C (Mark Christopher). “A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema.” 1993. Thesis, University of North Texas. Accessed August 08, 2020. https://digital.library.unt.edu/ark:/67531/metadc500353/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Huggins, Mark C (Mark Christopher). “A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema.” 1993. Web. 08 Aug 2020.

Vancouver:

Huggins MC(C. A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema. [Internet] [Thesis]. University of North Texas; 1993. [cited 2020 Aug 08]. Available from: https://digital.library.unt.edu/ark:/67531/metadc500353/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Huggins MC(C. A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema. [Thesis]. University of North Texas; 1993. Available from: https://digital.library.unt.edu/ark:/67531/metadc500353/

Not specified: Masters Thesis or Doctoral Dissertation