# construct a sequence of continuous functions satisfy these conditions

• Oct 18th 2008, 02:27 PM
szpengchao
construct a sequence of continuous functions satisfy these conditions
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.
• Oct 18th 2008, 02:58 PM
Plato
Which of these two do you mean: $\displaystyle \left( {\int\limits_{ - 1}^1 {f_n } } \right) not\to 0\,\mbox{ or } \,\left( {\forall n} \right)\left[ {\int\limits_{ - 1}^1 {f_n } \ne 0} \right]$?
• Oct 18th 2008, 02:59 PM
szpengchao
for all
for all n i think.
• Oct 18th 2008, 04:14 PM
Plato
That is almost the trivial case.
$\displaystyle f_n (x) = \left\{ {\begin{array}{rl} 0 & {x \notin \left( {\frac{{ - 1}} {n},\frac{1} {n}} \right)} \\ {x + \frac{1} {n}} & {x \in \left( {\frac{{ - 1}} {n},0} \right)} \\ { - x + \frac{1} {n}} & {x \in \left[ {0,\frac{1} {n}} \right)} \\ \end{array} } \right.$
• Oct 19th 2008, 03:46 AM
szpengchao
• Oct 19th 2008, 04:14 AM
Plato
O.K. Here is a start.
$\displaystyle f_1 \left( x \right) = \left\{ {\begin{array}{rl} {4\left( {x - \frac{1} {2}} \right) + 2} & {x \in \left( {0,\frac{1} {2}} \right]} \\ { - 4\left( {x - \frac{1} {2}} \right) + 2} & {x \in \left( {\frac{1} {2},1} \right]} \\ 0 & {\mbox{else }} \\ \end{array} } \right.$