I'm thinking of a particular example to show that
there isn't a sequence of continuous function on $[0,1]$ that converges pointwise to the function $f$ on $[0,1]$ defined by $f(x)=0$ if $x$ is rational and $f(x)=1$ if $x$ is irrational.
However, the only answer provided invoked Baire's theorem, which seems like unnecessary machinery. I'm wondering about a solution that only uses the definition of continuity and point-wise convergence.
This is a slightly different question. Is there any consensus on how I should propose further inquiry into the question?