March 23rd, 2018, 08:19 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Differentiable or not
Let $f(x) = x$ ; $x \in Q$ $0$; $x \notin Q$ The function at point $x = 0$ a) Is differentiable b) has the left derivative but not the right derivative c) has the right derivative but not the left derivative d) has neither the left derivative nor the right derivative Since both the functions are equal at $x = 0$, I think it is differentiable. Please help! Last edited by skipjack; March 26th, 2018 at 02:02 AM. 
March 23rd, 2018, 08:57 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 520 Thanks: 293 Math Focus: Dynamical systems, analytic function theory, numerics 
What is the definition of differentiability?

March 23rd, 2018, 09:51 PM  #3 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
March 23rd, 2018, 10:07 PM  #4 
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  
March 23rd, 2018, 10:45 PM  #5 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
March 24th, 2018, 12:48 AM  #6 
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  $f$ is the constant function that inputs any real number and always returns the value $1$. What is its derivative? What's its value at $x = 0$?
Last edited by Maschke; March 24th, 2018 at 12:57 AM. 
March 24th, 2018, 05:47 AM  #7 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
March 24th, 2018, 06:54 AM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
The derivative of a function at a point is completely independent of the value of the function at that point, so your "but f(0) = 1" is irrelevant. The function f(x) = 1 has derivative 0 at any x. But the function in the original post has value x for any rational x, 0 for any irrational x. The difference quotient $\displaystyle \frac{f(0+ h) f(0)}{h}$ is $\displaystyle \frac{0+ h 0}{h}= 1$ for h rational and $\displaystyle \frac{0 0}{h}= 0$ for h irrational. The limit, as h goes to 0, does not exist, so this function is not differentiable at x = 0. Last edited by skipjack; March 26th, 2018 at 02:05 AM. 
March 24th, 2018, 09:58 AM  #9 
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Can you graph $f(x) = 1$? You are confusing the definitions of continuity and differentiability. This example is so elementary that given your educational aspiration, you cannot afford to misunderstand it.

March 26th, 2018, 01:31 AM  #10  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
Last edited by skipjack; March 26th, 2018 at 02:05 AM.  

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