I don't think such a sequence exists.
I have a proof (given in class) that if the sequence is equicontinuous and converges point-wise (non-uniformly) for every rational on $[a,b]$ then it converges uniformly on all of $[a,b]$.
I also have a proof (written by me) that if a sequence of continuous functions defined on $[a,b]$ converges point-wise for every rational on $[a,b]$, then it converges.
EDIT: "uniformly" change to "pointwise"
on all of $[a,b]$.
But I was told that equicontinuity is essential. I'm having hard time seeing why. Can someone give me a counterexample to demonstrate the need for equicontinuity?
My understanding is at the level of Rudin's principles [...] chapters 1-6 and most of chapter 7.