$f_n(x):[0,1]\rightarrow [0,1]$ are continuous and converges to $f(x)$, then
$f$ is continuous.
Convergence is uniform on $[0,1]$
Convergence is uniform on $(0,1)$
None of above statement is true.
Well, for 1 take $f_n(x)=x^n$, $f(x)=0$ for $ x \in [0,1)$ and $f(x)=1$ at $x=1$ hence 1 is false. For 2 I can give the same counter example as of 1. I have no idea about 3. Please help.