Nope, i don't think there are any. Since if pointwise then
which would mean f is not bounded on [0, 1] (since ) thus it must not continuous on [0, 1].
I want to find a sequence (f_k)of continuous bounded functions on [0,1]
converging pointwise to a continuous limit, but such that
the sup-norm of f_k tends to infinity when k tends to infinity...
is there any?
What you need to do is to ensure that the function has a narrow spike somewhere. For example, define in the interval , in the interval , and in the remainder of the unit interval. Then as , for each (fixed) point x in the unit interval.
That is not true, because pointwise convergence does not imply convergence in the sup norm.