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

In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function ? which is the uniform limit of a sequence of sawtooth functions ?ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let ? = [0, 1] x [0, 1] and ?(?) be the Hausdorff metric space determined by ?. We define contraction maps ?₁ , ?₂ , ?₃ on ?. These maps define a contraction map ? on ?(?) via ?(?) = ?₁(?) ⋃ ?₂(?) ⋃ ?₃(?). The iteration under ? of the diagonal in ? defines a sequence of graphs of continuous functions ?ₙ. Since ? is a contraction map in the compact metric space ?(?), ? 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 ?. 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 ?[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 ?[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 ? 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 ? and ? denote the natural numbers and the real numbers, respectively.