4
$\begingroup$

A function $f:\Bbb{R}\to\Bbb{R}$ is defined by $f(x)=x$, if $x$ is rational; $\sin(x)$ if $x$ is irrational.

Show that $f$ is differentiable at $0$ and $f'(0)=1$.

Here I'm thinking to apply definition separately for rational and irrational.. and I'm getting same limit... But can we conclude the differentiability with this result??

$\endgroup$
3
$\begingroup$

Yes, the fact that this holds for rationals and irrationals separately (but consistently) does allow you to conclude differentiability. In particular, you are trying to show that the limit $$\lim_{h\rightarrow 0}\frac{f(h)}h=1$$ and you have that $f(x)$ at every point is equal to either $f_1(x)=x$ or $f_2(x)=\sin(x)$ at every point. You have $$\lim_{h\rightarrow 0}\frac{f_1(h)}h=1$$ $$\lim_{h\rightarrow 0}\frac{f_2(h)}h=1$$ Then, you can conclude that the original limit exists as follows: For any $\varepsilon$, you can choose $\delta_1$ and $\delta_2$ such that for any $|x|<\delta_1$ you have $\left|\frac{f_1(h)}h-1\right|<\varepsilon$ and for any $|x|<\delta_2$ you have $\left|\frac{f_2(h)}h-1\right|<\varepsilon$. This is just from definition of the latter two limits. Then, for any $|x|<\min(\delta_1,\delta_2)$ you get that $\left|\frac{f(h)}h-1\right|<\varepsilon$, since one of the two above inequalities will be applicable. Thus, $f(h)$ must have the same derivative as $f_1$ and $f_2$ where those two functions and their derivatives agree.

$\endgroup$
  • $\begingroup$ Thank You, Can this function be differentiable at any point x≠0? $\endgroup$ – user334155 May 1 '16 at 5:42
  • $\begingroup$ @user334155 No; it's not even continuous anywhere else. I edited to clarify this in my answer. $\endgroup$ – Milo Brandt May 1 '16 at 15:07
1
$\begingroup$

By definition we need to compute $\displaystyle \lim_{h\to 0}\frac{f(0+h)-f(0)}{h}=\displaystyle \lim_{h\to 0}\frac{f(h)-0}{h}=\lim_{h\to 0}\frac{f(h)}{h}$

Notice for any $x\in \mathbb{R}$ we have $|x|\geq|\sin(x)|$ (Easy to check using mean value theorem). That means for $h>0$, we have $\sin(h)\leq f(h)\leq h\Rightarrow \frac{\sin(h)}{h}\leq \frac{f(h)}{h}\leq 1$, by squeeze theorem $\displaystyle \lim_{h\to 0^{+}}\frac{f(h)}{h}=1$. When $h<0$ the inequality reverses and we can also use squeeze theorem to conclude $\displaystyle \lim_{h\to 0^{-}}\frac{f(h)}{h}=1$.

Therefore $\displaystyle \lim_{h\to 0}\frac{f(h)}{h}=1$, the function is differentiable at $0$ and the derivative is 1.

$\endgroup$
  • $\begingroup$ In case, f(x)= x² , x€Q ; 0, x€ R/Q. As x->0, f(x)/x ->0. But this is possible if 0≤{f(x)/x}≤x. How can we show??? $\endgroup$ – user334155 May 1 '16 at 6:02
  • $\begingroup$ Similarly, what in case, f(x)=0, if x=0 and x€R/Q ; 1/q³, if x=p/q, where p€Z, q€N and (p,q)=1 $\endgroup$ – user334155 May 1 '16 at 6:20
1
$\begingroup$

If you put $g(x)=f(x)-x$, then it suffices to show $g'(x)=0$.

If you are able to show that $|g(x)|\le x^2$ then you have $$-x \le \frac{g(x)}x \le x$$ which implies $g'(x)=\lim\limits_{x\to0}\frac{g(x)}x=0$ (using squeeze theorem).

To prove the inequality for $|g(x)|$ you may use some inequality for $\sin x$. Some such inequalities can be found on this site. For example, see this question: Prove that $x - \frac{x^3}{3!} < \sin x < x$ for all $x>0$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.