I am trying to find a simple example of a function $f:[0,1]\rightarrow\mathbb{R}$ which is a pointwise limit of continuous functions, but is not Riemann integrable. I know the classical example where we build some functions $F_1,F_2,\dots$ on a cantor-like set, and then define $f_n = F_1.F_2.\dots . F_n$, and so on. But I was thinking whether there is a simpler example, one that you could present to students with no experience in Measure Theory.

Any help is welcome.

  • 3
    $\begingroup$ You need a function with a set of discontinuities of positive measure. The construction in this Wikipedia page (see the characteristic function of the Cantor set; it is approximated by the continuous functions [...]) might be useful. Take a set of positive measure and apply the construction [...]. This yields a sequence of continuous functions converging to a function that is discontinuous precisely of the set you chose previously. $\endgroup$ – Giuseppe Negro Feb 28 '18 at 18:32

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