4
$\begingroup$

I saw a question on stackexchange earlier today about the function $f(x)=\begin{cases} 1&\text{if }x\in\mathbb{Q}\\ 0&\text{otherwise.} \end{cases}$

It reminded me of an old problem (that I don't think I solved), and I thought it was worth sharing with others. Let \begin{equation*} g(x)=\begin{cases} \frac{1}{q^j}&\text{if }x\in\mathbb{Q}\\ 0&\text{otherwise.} \end{cases} \end{equation*} For rational values of $x$, we have $x=\frac{p}{q}$ is in lowest terms. Additionally, $j> 1$.

1) One of the questions that was asked was to show the following: $g$ is differentiable for $x\in\mathbb{R}\setminus\mathbb{Q}$.

I don't think this is very difficult; however, I had trouble with the next one:

2) Show that $g$ is $k$-times differentiable for $x\in\mathbb{R}\setminus\mathbb{Q}$ and $k

How do we prove (2)? Is it even true?

For the second derivative, we take \begin{equation*} f''(x)=\lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2} \end{equation*}

I'm not totally sure how one might extend to higher derivatives, and a quick google search did not yield what I was looking for! I would be happy if there was a proof for $k=2$ in my question.

  • 0
    For a function to be at least $2$ times differentiable in a point it needs to have a first derivative in a neighborhood of that point. How else would you define a second derivative? – 2017-01-15
  • 0
    Use $\lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}$ for the second order derivative at an irrational point. – 2017-01-15
  • 0
    I believe $g(x)$ is not differentiable at $x=\sum_{n\geq 1} 10^{-n!}$ – 2017-01-15
  • 0
    It seems to me as though it would be highly differentiable at liouville numbers. Wouldn't nearby rationals have large denominators? – 2017-01-15
  • 1
    Liouville numbers have very good rational approximations with small denominators. For the $x$ I mentioned, take $h=-\sum_{n>N} 10^{-n!}$ for some $N$. Then $h\approx 10^{-(N+1)!}$, but $g(x+h)=10^{-j\cdot N!}$, so $h$ is much smaller that $g(x+h)-g(x)$. – 2017-01-16
  • 0
    That's interesting. I suppose my memory of the problem changed some details. Perhaps the 'easier' case was simply continuity or Riemann integrability. – 2017-01-16

0 Answers 0