# construct a sequence of continuous functions satisfy these conditions

• Oct 18th 2008, 03:27 PM
szpengchao
construct a sequence of continuous functions satisfy these conditions
construct a sequence $(f_{n})$ of continuous real-valued functions on [-1,1] converging pointwise to the zero function but with $\int_{-1}^{1}{f_{n}}$ not equal 0.
• Oct 18th 2008, 03:58 PM
Plato
Which of these two do you mean: $\left( {\int\limits_{ - 1}^1 {f_n } } \right) not\to 0\,\mbox{ or } \,\left( {\forall n} \right)\left[ {\int\limits_{ - 1}^1 {f_n } \ne 0} \right]$?
• Oct 18th 2008, 03:59 PM
szpengchao
for all
for all n i think.
• Oct 18th 2008, 05:14 PM
Plato
That is almost the trivial case.
$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.$
• Oct 19th 2008, 04:46 AM
szpengchao
$f_1 \left( x \right) = \left\{ {\begin{array}{rl}