| Summary: | This thesis is, in the most general sense, about analysis of real and complex-valued functions on the interval [0,1]. In every instance, we can interpret the following discussions as explorations of the regularity properties of functions. In this, we touch on one can view a number of extremes of the meaning of regularity. On one hand, we will discuss the question of minimal sufficient conditions for the existence of derivatives in the same vein of Rademacher and Stepanov. In particular, we will examine the differentiability properties of continuous functions defined on Euclidean space. On the other hand, we will investigate the stability of the regularity of special solutions to certain dispersive PDEs assuming an arbitrarily high amount of Sobolev regularity. In between these two subjects lies an analysis of pointwise convergence of trigonometric polynomials to integrable functions.
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