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March 23rd, 2018, 07:19 PM   #1
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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.
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March 23rd, 2018, 07:57 PM   #2
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What is the definition of differentiability?
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March 23rd, 2018, 08:51 PM   #3
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What is the definition of differentiability?
If the left hand derivative is equal to the right hand derivative equals to $f(a)$ then it is differentiable.
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March 23rd, 2018, 09:07 PM   #4
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Originally Posted by Lalitha183 View Post
If the left hand derivative is equal to the right hand derivative equals to $f(a)$ then it is differentiable.
If $f(x) = 1$ then the left and right hand derivatives at $0$ are $0$, but $f(0)$ = 1, right?
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March 23rd, 2018, 09:45 PM   #5
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Quote:
Originally Posted by Maschke View Post
If $f(x) = 1$ then the left and right hand derivatives at $0$ are $0$, but $f(0)$ = 1, right?
I don't understand what does it mean by $f(x) = 1$ ? And how will $f(0) = 1 $ ?
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March 23rd, 2018, 11:48 PM   #6
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Quote:
Originally Posted by Lalitha183 View Post
I don't understand what does it mean by $f(x) = 1$ ? And how will $f(0) = 1 $ ?
$f$ 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.
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March 24th, 2018, 04:47 AM   #7
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Quote:
Originally Posted by Maschke View Post
$f$ is the constant function that inputs any real number and always returns the value $1$. What is its derivative? What's its value at $x = 0$?
The Left Hand Derivative = Right Hand Derivative = $0$. But $f(0) = 1$. So the function is not differentiable at $x=0$.
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March 24th, 2018, 05:54 AM   #8
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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.
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March 24th, 2018, 08:58 AM   #9
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Quote:
Originally Posted by Lalitha183 View Post
The Left Hand Derivative = Right Hand Derivative = $0$. But $f(0) = 1$. So the function is not differentiable at $x=0$.
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.
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March 26th, 2018, 12:31 AM   #10
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Quote:
Originally Posted by Country Boy View Post
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.
Which one do we take as RHD & LHD? Because to my question, answers are like is differentiable/ has L.H.D but not R.H.D/ has R.H.D but not L.H.D/ Neither L.H.D nor R.H.D.

Last edited by skipjack; March 26th, 2018 at 01:05 AM.
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