= Solution
{c}
Let $f_n$ be the <continuous> triangular bump
$$
f_n(x)=\begin{cases}
n^2x,&0\leq x\leq1/n,\\
n(2-nx),&1/n\leq x\leq2/n,\\
0,&2/n\leq x\leq1,
\end{cases}\qquad n\geq2,
$$
and set $f_1=0$. At $x=0$ every term is zero, and each fixed $x>0$ eventually lies outside the support, so $f_n\to0$ <pointwise>. The triangle has base $2/n$ and height $n$, hence
$$
\boxed{\int_0^1 f_n(x)\,dx=1\quad(n\geq2),\qquad \int_0^1 0\,dx=0.}
$$
This <concentrating triangular bump> shows that <pointwise convergence> does not preserve integrals, even for <continuous functions> on a <compact interval>.
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