Why can't Dirichlet higher numbers be expressed as limit functions of continuous functions

Why can't Dirichlet higher numbers be expressed as limit functions of continuous functions

This is a very important theorem in the theory of real variable function. It reveals the relationship between almost everywhere convergence and uniform convergence of function sequence. In mathematical analysis, it is usually called quasi uniform convergence. Roughly describe this theorem, that is to say, almost everywhere convergence (or convergence) of function sequence on a certain set, most of which are

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