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Math Help - construct a sequence of continuous functions satisfy these conditions

  1. #1
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    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.
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  2. #2
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    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]?
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  3. #3
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    for all

    for all n i think.
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  4. #4
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    That is almost the trivial case.
    f_n (x) = \left\{ {\begin{array}{rl}<br />
   0 & {x \notin \left( {\frac{{ - 1}}<br />
{n},\frac{1}<br />
{n}} \right)}  \\<br />
   {x + \frac{1}<br />
{n}} & {x \in \left( {\frac{{ - 1}}<br />
{n},0} \right)}  \\<br />
   { - x + \frac{1}<br />
{n}} & {x \in \left[ {0,\frac{1}<br />
{n}} \right)}  \\<br /> <br />
 \end{array} } \right.
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  5. #5
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    what about

    what about the other case?
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  6. #6
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    O.K. Here is a start.
    f_1 \left( x \right) = \left\{ {\begin{array}{rl}<br />
   {4\left( {x - \frac{1}<br />
{2}} \right) + 2} & {x \in \left( {0,\frac{1}<br />
{2}} \right]}  \\<br />
   { - 4\left( {x - \frac{1}<br />
{2}} \right) + 2} & {x \in \left( {\frac{1}<br />
{2},1} \right]}  \\<br />
   0 & {\mbox{else     }}  \\ \end{array} } \right.
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