A discussion upon the definitions of integrable functions using Riemann and Darboux sums - equivalence etc.Several proofs of Bolzano's theorem for continuous functions - related to the previous one.through suprema or using Cantor's principle or using Bolzano-Weierstass's theorem orthrough Heine-Borel theorem etc. The critical comparison of the several ways in which one may express the completeness axiom - e.g.Some sample topics related to calculus could be the following: This may be way off the usual written examination, it makes it, however, necessary for the students to process the knowledge they have obtained conceptually and not only "copy-and-paste" from the internet or each other. For instance, there could be created a pool of project topics which will be randomly assigned to students. Well, one idea would be that of shifting to evaluating students through projects.
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