The sequence
\( f_n(x)= \begin{cases}n^2 x, & 0 \leq x \leq \frac{1}{n} \\ -n^2 x+2 n, & \frac{1}{n} \leq x \leq \frac{2}{n} \\ 0, & \frac{2}{n} \leq x \leq 1\end{cases} \)
1
not pointwise convergent
2
the point wise limit is f(x) = x
3
\( \lim _{n \rightarrow \infty} \int_0^1 f_n(x) d x=\int_0^1 f(x) d x\)
4
\(\lim _{n \rightarrow \infty} \int_0^1 f_n(x) d x \neq \int_0^1 f(x) d x\)