Real Analysis – Investigating uniform convergence of a sequence of functions to a function g

real-analysis

Question is that find a sequence ($f_n$) of continuously differentiable real functions defined on $[0,1]$ converges uniformly to a differentiable function ($g$) and ($f_n'$) converge pointwise to a function that is not ($g'$). I am trying to find this function. I think that $\frac{x}{1+x^2n^2}$ converges uniformly to zero on [0,1] and derivate $\frac{1-n^2x^2}{(1+x^2n^2)^2}$ converges pointwise to 1 at zero and 0 for all other points. Is my logic correct? Any help will be appreciated.Thanks

Best Answer

This one is a bit old now but many are redirected here when they ask a similar question. So perhaps a definitive answer might be welcome.

Q1. Suppose that $\{f_n\}$ are continuously differentiable functions converging uniformly to a differentiable function $f$ on $[0,1]$. Suppose that $\{f_n'\}$ converges pointwise there to a function $g$.

What can be said about the set $M= \{x\in [0,1]: f'(x)\not = g(x)\}$?

and

Q2. Same question but drop the "continuously differentiable" assumption and just assume that each $f_n$ is differentiable.

The answer one sees here and very often is that $M$ can be nonempty. Specific examples abound where $M$ contains a single point. If your ambitions are that tiny then don't read any further.

There is a definitive answer to these questions though. Many posters here would no doubt like to know this.

There are two aspects to questions like this. One wants to know what type of set $M$ must be. In particular it is expected that $M$ should be some simple kind of Borel set. Just how complicated can it be?

Secondly just how large a set might $M$ be. Dense? Uncountable? Etc?

A young graduate student asked these questions not so long ago. When Darji was a graduate student a little more than 30 years ago he asked this question of his advisor (Jack Brown) and he didn't know. That led to Darji's thesis topic.

Darji, Udayan B. Limits of differentiable functions. Proc. Amer. Math. Soc. 124 (1996), no. 1, 129-134.

Abstract. Suppose that $\{f_n\}$ is a sequence of differentiable functions defined on $[0,1]$ which converges uniformly to some differentiable function $f$, and $f_n'$ converges pointwise to some function $g$. Let $M= \{x: f'(x)\not = g(x)\}$. In this paper we characterize such sets $M$ under various hypotheses. It follows from one of our characterizations that $M$ can be the entire interval $ [0,1]$.

Here are the answers that this paper provides:

  1. The set $M$ in the continuously differentiable case must be of type $\cal F_\sigma$.

  2. Moreover, any set $M$ that is of type $\cal F_\sigma$ and is a nowhere dense subset of $[0,1]$ can appear in the continuously differentiable case.

  3. In particular then $M$ could be any Cantor-like subset of $[0,1]$.

  4. For the more general case (where the $f_n$ are just assumed to be differentiable) the set $M$ must be of type $\cal G_{\delta\sigma}$.

  5. Surprisingly (to me anyway) given any set $M$ of type $\cal G_{\delta\sigma}$ there is an example where $M= \{x\in [0,1]: f'(x)\not = g(x)\}$.

  6. That means then, that $M$ could be all of $[0,1]$.

If #3 and #5 don't quite surprise you then you are too jaded to be reading answers on StackExchange.

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