# Sequence of continuous functions $(f_n)$ that converges to the zero function and $\int_0^1 f_n(x)dx$ increases without a bound

Is it possible to find:

Sequence of continuous functions $f_n:[0,1]\rightarrow \mathbb{R}$ that converges to the zero function and such that sequence $\int_0^1 f_1(x)dx, \int_0^1 f_2(x)dx,\ldots$ increases without a bound

I think it's quite easy. Just define $f_n$ in the following way:

Uniformly convergent sequence of differentiable functions $f_n : (0,1)\rightarrow \mathbb{R}$ such that the sequence $f_1 ', f_2',\ldots$ does not converge.

Here I have trouble.

Convergent sequence of Riemann integrable functions $f_n : [0,1]\rightarrow \mathbb{R}$ whose limit function IS NOT Riemann integrable.

I know the example that uses charactersitic function of rationals. But is it possible to give another example?

• Your first example doesn't converge at $0$; but that's easily fixed ( /\ ). – David Mitra Dec 14 '15 at 16:35
• cHere I wrote up some continuous functions that have constant integral but converge pointwise to zero: math.stackexchange.com/questions/1553955/… If you multiply the function $f_n$ by n pointwise convergence to zero will still be given but the integral will diverge. – s.harp Dec 14 '15 at 16:35
• Just make the graph look like what I wrote at the end of my first comment. – David Mitra Dec 14 '15 at 16:52
• The triangle will have vertices at the origin, $(1/(2n),2n^2)$, and $(1/n,0)$. Its area will be ${1\over 2}\cdot 2n^2\cdot{1\over n}=n$. – David Mitra Dec 14 '15 at 17:05
• And it will still converge to $0$, of course. – David Mitra Dec 14 '15 at 17:07

(second question) $f_n(x)=\frac1n\sin(n^2x)$
(third question) $f_n(x)=1/x$ on $[1/n,1]$ and $n^2x$ on $[0,1/n].$
• (i) has finite integral on $[0,1]$, (ii) is not continuous. – s.harp Dec 14 '15 at 16:46