This is the function: $$f(x)=\begin{cases}x& \text{if $x$ is rational}\\0 &\text{if $x$ is irrational}\end{cases}$$

My attempt:

It's easy to verify that $f$ is continuous at $x=0$ using the sequential definition of continuity. I claim that $f$ is not differentiable at $x=0$. Assume the contrary and let $f'(0)=L$. Now, we pick an $\varepsilon$ such that $0<\varepsilon < |L|$. For this choice of $\varepsilon$ there is a $\delta >0$ such that if $0<|x-0|<\delta$ then we have $\left| \frac{f(x)-f(0)}{x-0} -L\right| < \varepsilon $. Now, pick $x' \in \mathbb{R}\setminus\mathbb{Q}$ with $0<|x'| <\delta$. Then we have $\left| \frac{f(x')-f(0)}{x'-0} -L\right| = |L| > \varepsilon$. A contradiction!

Is this proof correct?


EDIT: As @DanielWainfleet pointed out in the comments, your proof does not apply in the case $L=0$. The proof basically goes exactly like the one you have written: first you choose an $0<\varepsilon<1$ small enough, then you take $\delta$ as you please and look at the difference quotient for any $|x|<\delta$. If $f$ was differentiable, then it should hold that $$\left|\frac{f(x)-f(0)}{x-0}\right|<\varepsilon$$ but by density of the irrationals, you can find an irrational $x$ such that $|x|<\delta$ and for that one $$\left|\frac{f(x)-f(0)}{x-0}\right|=\left|\frac{x-0}{x-0}\right|=1>\varepsilon$$ contradicting the assumption.

Another working approach is showing that approaching $0$ via irrationals and via rationals leads to different difference quotients:

Indeed :

  • choose a sequence of irrational numbers ($a_n=\sqrt{2}/n$) tending to zero. Then $\frac{f(a_n)-0}{a_n-0}=0$ for all $n$
  • choose a sequence of rational numbers ($a_n=1/n$) tending to zero. Then $\frac{f(a_n)-0}{a_n-0}=1$ for all $n$
  • $\begingroup$ The flaw in the proposer's proof is assuming that if $L$ exists then $L\ne 0.$ If $L=0$ then we cannot take any $\epsilon$ such that $0<\epsilon <|L|.$ $\endgroup$ – DanielWainfleet Nov 17 '18 at 7:07
  • $\begingroup$ @DanielWainfleet You are indeed correct. My bad for missing it! I'll add the correction in my answer $\endgroup$ – b00n heT Nov 17 '18 at 8:08
  • $\begingroup$ Much of performance-magic depends on how easily we make or overlook unwarranted assumptions. $\endgroup$ – DanielWainfleet Nov 17 '18 at 11:34

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