construct a sequence $\displaystyle (f_{n})$ of continuous real-valued functions on [-1,1] converging pointwise to the zero function but with $\displaystyle \int_{-1}^{1}{f_{n}}$ not equal 0.
construct a sequence $\displaystyle (f_{n})$ of continuous real-valued functions on [-1,1] converging pointwise to the zero function but with $\displaystyle \int_{-1}^{1}{f_{n}}$ not equal 0.
That is almost the trivial case.
$\displaystyle f_n (x) = \left\{ {\begin{array}{rl}
0 & {x \notin \left( {\frac{{ - 1}}
{n},\frac{1}
{n}} \right)} \\
{x + \frac{1}
{n}} & {x \in \left( {\frac{{ - 1}}
{n},0} \right)} \\
{ - x + \frac{1}
{n}} & {x \in \left[ {0,\frac{1}
{n}} \right)} \\
\end{array} } \right.$
O.K. Here is a start.
$\displaystyle f_1 \left( x \right) = \left\{ {\begin{array}{rl}
{4\left( {x - \frac{1}
{2}} \right) + 2} & {x \in \left( {0,\frac{1}
{2}} \right]} \\
{ - 4\left( {x - \frac{1}
{2}} \right) + 2} & {x \in \left( {\frac{1}
{2},1} \right]} \\
0 & {\mbox{else }} \\ \end{array} } \right.$