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This book§presents a comprehensive selection from the extensive literature on discrete§and integral inequalities. It is the first part of a two-volume§presentation. Integral§and discrete inequalities are a very important tool in classical analysis with at least two major applications,§namely ODEs (Ordinary Differential Equations) and PDEs (Partial Differential§Equations). The compilation includes integral, differential and difference§inequalities, which play a crucial role in establishing the well-posedness of§the related equations.§§Volume II§(Chapter 5-8) focuses on nonlinear one-dimensional inequalities and nonlinear§multi-dimensional inequalities.§