I'm studying for my analysis functions and we've been working on sequences on functions. One of the practice questions we were given asks us to find a sequence of continuous functions $f_n : [0,1] \to\ \mathbb R$ that converges pointwise to a continuous function and $$\lim\limits_{n\to\infty} \int_{0}^{1} f_n(x)dx ≠ \int_{0}^{1} f(x)dx$$

We are also asked to find a sequence of continuous functions with all the same conditions, except converging uniformly to a continuous function.

I really have no idea where to begin with this. Any ideas what these sequences of functions should be? Thanks in advance.

  • 2
    $\begingroup$ For the first question consider a sequence of functions $f_n$ with a spike on $[0, 2/n]$, with peak in $1/n$ such that the integral is always $=1$ say. For the second one have another look at the theorems you have learned about uniform convergence and integration. $\endgroup$ – Thomas Apr 6 '14 at 15:12
  • $\begingroup$ Do you know any theorems about integrals of a uniformly convergent sequence of functions? $\endgroup$ – Ted Shifrin Apr 6 '14 at 15:13
  • $\begingroup$ @TedShifrin Looking at the theorem in my book about if (f_n) converges uniformly to f, then f is Riemann integrable. But it also says that in that case, lim of the integral of (f_n) IS equal to the integral of f. Which isn't supposed to hold true for this question. So I'm not sure I understand where you're going with that $\endgroup$ – user114014 Apr 6 '14 at 15:20
  • $\begingroup$ @Thomas, are you saying that the "spike" on [0, 2/n] and the "peak" at 1/n are the same thing? Can you maybe give me an example of that kind of function? I find this explanation unclear. $\endgroup$ – user114014 Apr 6 '14 at 15:21
  • $\begingroup$ Well, you've answered the second question. There can be no such example. $\endgroup$ – Ted Shifrin Apr 6 '14 at 15:27

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