# Analysis Constructing a Sequence

I'm looking for a sequence of functions that is continuous and absolutely integrable, but pointwise divergent for every $z$ $\in [0,1]$.

In other words,

$\int_0^1 |f_n(z)| dz \rightarrow 0$ as $n \rightarrow \infty$, but $(f_n(z))_n$ pointwise diverges for every $z$ $\in [0,1]$.

I'm having no luck coming up with an example, and would appreciate any sort of insight on how to go about constructing such a sequence.

• What do you mean by "diverges"? Do you just mean that $\lim_{n\to\infty} f_n(z)$ doesn't exist, or that $\lim_{n\to\infty} f_n(z)=+\infty$? – Math1000 May 11 '15 at 2:25
• I guess diverges would mean "not converge" to a finite number . @Math1000 – user99914 May 11 '15 at 2:28
• @Math1000 John is right, that's what I mean. – Eddie May 11 '15 at 2:31

For integers $$n > 0$$ and $$0 \leq m < n$$, Define $$f_n^m(z) = 1$$ if $$z \in [m/n,(m+1)/n]$$ (call this interval $$I_n^m$$), and $$0$$ otherwise. Then "smooth the edges" (to make them continuous) of these functions so that they're not quite step functions, and arrange them in the sequence $$f_1^0,f_2^0,f_2^1,f_3^0,f_3^1,f_3^2,f_4^0,f_4^1,f_4^2,f_4^3,\dots$$
Each of these diverges pointwise, because for any $$x \in [0,1]$$, you should be able to find two subsequences of $${f_n^m(x)}$$, one that converges to $$0$$ and another that converges to $$1$$. But the integrals get smaller and smaller since $$\lim_{n\to\infty}\mu(I_n^m) = \lim_{n\to\infty} \frac 1 n = 0,$$ where $$\mu([a,b]) = b-a$$. Not quite a precise answer, but if you work out the details it should work.
Edit: If $$J_n^m = \text{Interior}(I_n^m)$$, then instead of taking step functions and smoothing them, you can just choose continuous $$f_n^m$$ such that $$f_n^m = 0$$ outside of $$J_n^m$$, and $$f_n^m \equiv 1$$ on $$I_n^m \setminus (\text{tiny boundary interval})$$ and the result should hold. Examples include triangles, Gaussians, trapezoids, etc.
• Just requiring $f^m_n > 0$ seems to not be enough as these values may converge to $0$ as $m$ and $n$ get bigger. – JKEG Sep 13 '19 at 14:32