The fact that $\chi_{\mathbb Q}$ is not a pointwise limit of continuous functions has been discussed a bunch of times, but the answers are scattered.
I posted an answer to #77307 and then found #64677. Tried to flag the latter as a duplicate of the former, but was foiled by the lack of upvote to the answer.
Oh, and thanks :). (Sadly, it is unlikely that it gets (much) more votes, while I like the argument and this type of thing is usually glossed over as trivial.)
