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Example 8.1.8 (e): Pointwise Convergence does not preserve Integration

Find a pointwise convergent sequence of Riemann-integrable functions whose limit is Riemann-integrable but
fn(x) dx # fn(x) dx = f(x) dx

This is perhaps a little tricky, but we could try one of the function sequences we discussed earlier:

fn(x) = max(n - n2 |x - 1/n|, 0)

where x [ 0, 1 ] (you knew this function would play a bigger role, didn't you). Recall that each member of this family is piecewise linear, so another way to write fn is:

Each fn is continuous, hence Riemann integrable. The pointwise limit f is zero, which is again integrable. But when we integrate:

    
    
for all n. On the other hand, the integral of the limit function is clearly zero so that:
1 = fn(x) dx # f(x) dx = 0
By the way, if we consider the integral as area under a function it is immediately clear from the graph that the fn's integrate to 1 ...
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