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Define $f:\mathbb R\rightarrow \mathbb R $ by

$$ f(x)= \begin{cases} 0&,x\in \mathbb R \setminus \mathbb Q\\1&,x\in \mathbb Q \end{cases} $$

Let $x\in \mathbb R \setminus \mathbb Q$.

I was asked to calculate Dini derivatives of $f$ at $x$, i.e. $(D^+f)(x), (D_+f)(x), (D^-f)(x),$ and $(D_-f)(x)$. I obtained $(D_+f)(x)=0$ (which I'm not entirely sure is right) so far, and I can't seem to proceed further.

I can kind of guess that $(D^+f)(x)=∞ $ but I don't know how to show my working to obtain this. I think if I know how to compute the first two derivatives, the rest would be very much similar. Can someone please help? Thank you.

epsilonfox
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1 Answers1

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By definition

$(D^+f)(x)=\limsup_{h\to 0^+}\frac{f(x)+f(x+h)}{h}.$

Since $f(x)=0$,

$(D^+f)(x)=\limsup_{h\to 0^+}\frac{f(x+h)}{h}.$

Now, if you take a sequence $h_n\to 0$ of positive real numbers such that $x+h_n$ is rational (you can do it because there are rational numbers arbitrarily close to $x$), then

$\lim_{h_n\to 0^+}\frac{f(x+h_n)}{h_n}=\lim_{h_n\to 0^+}\frac{1}{h_n}=+\infty$

so your guess is right. You can try something similar for the other cases.

Javi
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