The Development Of Everywhere Continuous, Nowhere Differentiable Functions

While researching everywhere continuous, nowhere differentiable functions, one would find a variety of papers and different strategies proving the majority of continuous functions to be nowhere differentiable. This is a sharp contrast to the beliefs of late eighteenth and early nineteenth century mathematicians. It was not only falsely believed that all continuous functions are differentiable except for at some isolated points, but mathematicians that sought to solidify analysis and the definition of continuity faced strong opposition from their peers. Despite this opposition, many scholars became invested in formalizing mathematical logic and expanding on the ideas of the pioneers of pathological functions; like Weierstrass and Bolzano [1]. Functions like the Bolzano function and Weierstrass function, whose continuity and nowhere differentiability proofs are included in this paper, inspired the discovery of series and fractals with these same properties. This paper serves as a timeline of some of the most significant discoveries of everywhere continuous, nowhere differentiable functions, from the first publication of one in the late nineteenth century, to modern examples like Liu Wen’s function in 2002.