In this diploma thesis, we are interested in understanding which subsets of real numbers can be sets of discontinuity of a real function of one variable. We show that any set of discontinuity is a countable union of closed sets, which, for example, excludes the possibility of an existence of a real function that is discontinuous precisely at irrational numbers. This is shown as an application of the Baire category theorem. In the last part of the thesis we show that the pointwise limit of a sequence of continuous functions is always continuous on a large subset of real numbers.
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