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I want to give an example of a sequence of functions $f_1 \dots f_n$ that converges with respect to the metric $d(f,g) = \int_a^b |f(x) - g(x)| dx$ but does not converge pointwise.

I'm thinking of a function $f_n$ that is piecewise triangle, whose area converges to some constant function, but doesn't converge pointwise.

I just can't manage to formalize it.

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    Think of the most basic functions you know first. It helps. See my answer. =)2012-04-15
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    There is a standard example for this, think about the unit interval with a square function that slides back and fourth with the squares area going to 0 by means of it's width shrinking (keep height same).2012-04-15
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    Bleh, I misread your question. Sorry for the silly comment/answer. I've been computing differential equations for too long for one of my exams, and I began entering in robot mode / stop thinking..2012-04-15
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    I think the whole idea is just to move the area around fast enough so that pointwise convergence is not obtained.2012-04-15
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    I edited my answer. Take a look, it should be fine now =)2012-04-15

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