March 23rd, 2018, 07:19 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 01:02 AM. 
March 23rd, 2018, 07:57 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 377 Thanks: 205 Math Focus: Dynamical systems, analytic function theory, numerics 
What is the definition of differentiability?

March 23rd, 2018, 08:51 PM  #3 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
March 23rd, 2018, 09:07 PM  #4 
Senior Member Joined: Aug 2012 Posts: 1,887 Thanks: 525  
March 23rd, 2018, 09:45 PM  #5 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
March 23rd, 2018, 11:48 PM  #6 
Senior Member Joined: Aug 2012 Posts: 1,887 Thanks: 525  $f$ 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 23rd, 2018 at 11:57 PM. 
March 24th, 2018, 04:47 AM  #7 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
March 24th, 2018, 05:54 AM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,159 Thanks: 866 
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 01:05 AM. 
March 24th, 2018, 08:58 AM  #9 
Senior Member Joined: Aug 2012 Posts: 1,887 Thanks: 525  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, 12:31 AM  #10  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
Last edited by skipjack; March 26th, 2018 at 01:05 AM.  

Tags 
differentiable 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Differentiable Map  ZMD  Calculus  0  April 23rd, 2017 05:23 AM 
is this differentiable?  Omnipotent  Calculus  0  October 4th, 2014 01:46 PM 
differentiable at x_0  alphabeta89  Real Analysis  4  January 28th, 2012 01:56 AM 
differentiable at x_0  alphabeta89  Real Analysis  0  January 18th, 2012 02:52 AM 
Is f differentiable  SyNtHeSiS  Calculus  2  April 29th, 2010 03:08 PM 