8.1. Pointwise Convergence
Example 8.1.8 (e): Pointwise Convergence does not preserve Integration
Find a pointwise convergent sequence of Riemannintegrable functions whose
limit is Riemannintegrable but
f_{n}(x) dx # f_{n}(x) dx = f(x) dx
This is perhaps a little tricky, but we could try one of the function sequences we discussed earlier: f_{n}(x) = max(n  n^{2} 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 f_{n} is: Each f_{n} 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 = f_{n}(x) dx # f(x) dx = 0By the way, if we consider the integral as area under a function it is immediately clear from the graph that the f_{n}'s integrate to 1 ...