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find $\{f_n \}$ a sequence of continuous functions on $\mathbb{R}$ such that $0 \leq f_n(x) \leq 1$ and $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx =0$, but $f_n(x)$ does not converge anywhere on $[0,1]$.

Everytime I am thinking with a sequence, I come with one satisfies some of the conditions but violate one condition. For example, $f_n(x)=x^n$ is continuous sequence, and $f_n \rightarrow 0$ so limit the integral is zero, but it is pointwise convergent. , I tried to create a function similar to Dirchlet function, as they not converge pointwise, but they converge on part of the domain and they are not continuous.

math_for_ever
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1 Answers1

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Rather than give formulas, I will describe the graphs of the functions, which are piecewise linear. Think of a “discretely moving wave”, where after the wave is done moving, we switch to a “thinner” wave.

Let $f_1$ go from $(0,1)$ to $(\frac{1}{2},0)$ to $(1,0)$. Let $f_2$ go from $(0,0)$ to $\frac{1}{2},1)$ to $(1,0)$. Let $f_3$ go from $(0,0)$ to $\frac{1}{2},0)$ to $(1,1)$.

Then let $f_4$ go from $(0,1)$ to $(\frac{1}{4},0)$ to $(1,0)$. Let $f_5$ go from $(0,0)$ to $(\frac{1}{4},1)$ to $(\frac{1}{2},0)$ to $(1,0)$. Then $f_6$ from $(0,0)$ to $(\frac{1}{4},0)$ to $(\frac{1}{2},1)$ to $(\frac{3}{4},0)$ to $(1,0)$. Then $f_7$ go from $(0,0)$ to $(\frac{1}{2},0)$ to $(\frac{3}{4},1)$ to $(1,0)$. Then $f_8$ from $(0,0)$ to $(\frac{3}{4},0)$ to $(1,1)$.

Next do the same thing with the partition $0\lt \frac{1}{8}\lt\frac{1}{4}\lt\frac{3}{8}\lt \frac{1}{2}\lt \frac{5}{8}\lt\frac{3}{4}\lt\frac{7}{8}\lt 1$. Then partitioning $[0,1]$ into $16$ equal subintervals, etc.

Each new batch of functions has a smaller integral, all positive, but converging to $0$. The functions are all continuous. And $f_n(x)$ does not converge for any $x$ because you can always find arbitrarily large values of $n$ where $f_n(x)=0$ and values where $f_n(x)$ is very close to $1$ (by approximating $x$ with a rational with denominator a power of $2$).

Arturo Magidin
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  • I am curious to ask if these functions have a certain pattern or compact formula!! – math_for_ever Mar 23 '21 at 23:58
  • It is really a clever example! – math_for_ever Mar 23 '21 at 23:58
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    @math_for_ever: I’m sure we could come up with a relatively succint formula in terms of the nearest power of $2$ or something, but I don’t think there’s much point in expending the effort... – Arturo Magidin Mar 23 '21 at 23:59
  • you are right! Does this sequence has a name? – math_for_ever Mar 24 '21 at 00:11
  • @math_for_ever: I am not aware of one, though that doesn’t mean it, or something similar, does not have one. – Arturo Magidin Mar 24 '21 at 00:12
  • Are these continuous functions??? $f_2$ and $f_3$ are continuous at endpoints as $f_n$ are supposed to be continuous on $\mathbb{R}$.. – math_for_ever Mar 24 '21 at 01:19
  • @math_for_ever: Huh? I assumed continuous on $[0,1]$, but if you need them continuous on all of $\mathbb{R}$ just drop them to $0$ when needed with a similar line. What happens outside of $[0,1]$ is irrelevant to the properties of having the integrals go to $0$ and not converging pointwise inside $[0,1]$. – Arturo Magidin Mar 24 '21 at 01:25
  • @math_for_ever: if anyone is interested, i've plotted the very nice sequence in geogebra. if you let $n=0$ and let $j$ run you will see the first wave pass, with $n=1$ you will see the second, and so on. – Alan Mar 24 '21 at 05:19
  • @Alan, yes, I am interested. – math_for_ever Mar 24 '21 at 14:29
  • @math_for_ever: if you want the sequence to be continuous in $\mathbb{R}$ by doing what Arturo suggested, consider the sequence described by $k$ instead of $F$, or enlarge the domain of $h$ a bit. – Alan Mar 24 '21 at 14:57
  • @math_for_ever: This sequence is a continuous version of the typewriter sequence. https://math.stackexchange.com/questions/1412091/the-typewriter-sequence/1412099 – Alex Ortiz May 18 '21 at 17:12
  • @ArturoMagidin could you tell me if we can choose a $f_{n_k}$ subsequence for which $\lim_{k \rightarrow \infty} f_{n_k} $ exists a.e.? –  May 21 '21 at 14:02
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    @fourelements the subsequence consisting of the very first wave of each size converges pointwise to $\chi_{{x=0}}$. – Alan Jul 11 '21 at 23:53