Skip to main content
1 of 3
Robert Israel
  • 54.2k
  • 1
  • 76
  • 152

Let $\Delta$ be the rationals in $[0,1]$. If your condition is satisfied, in particular $f_n$ converges pointwise to $\chi_\Delta$. Let $C_n = \{x \in [0,1]: |f_n(x) - \chi_\Delta(x)|\le 1/3\}$. 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.

Robert Israel
  • 54.2k
  • 1
  • 76
  • 152