(EDITED)
Let $\Delta$ be the rationals in $[0,1]$.  If your condition is satisfied, in particular $f_n$ converges pointwise to $\chi_\Delta$.  Let $C_n = \{x \in [0,1]: \forall m > n, \;|f_n(x) - f_m(x)|\le 1/3\}$.  Then $C_n$ are closed and their union is $[0,1]$.  By the Baire category theorem some $C_n$ has nonempty interior.  But this is impossible since $f_n$ is continuous and both $\Delta$ and its complement are dense.