In this master's thesis we first presents the Lebesgue measure, which is introduced through an outer measure satisfying the condition of countable additivity. We also discuss the properties of Lebesgue measurable sets, focusing on step functions and simple functions that can be used to define the Riemann and Lebesgue integrals. Next we discuss the properties of the Riemann integral and introduce the Lebesgue integral, while referring to the properties of the measure and measurable functions. The main part of the master’s thesis revolves around some classes of continuous functions, especially almost everywhere differentiable functions, functions with bounded variation and absolutely continuous functions. It also presents nowhere differentiable functions, monotonic functions and the Lipschitz functions. Furthermore, we discuss the properties of these individual classes of functions, including their Riemann and Lebesgue integrability. The final part focuses on Lp spaces where we show that they are normed and complete vector spaces and that the class of simple functions is, in fact, a dense subspace of an Lp space.
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