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# If f(x) is continuous on an open interval, it is di

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If f(x) is continuous on an open interval, it is di erentiable there.

Guest Sep 3, 2014

### Best Answer

#5
+92788
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As one of my calculus teachers would have said.....differentiability implies continuity, but continuity does not imply differentiability.....

To see why, look at the absolute value function at x = 0. Note that, it is continuous at that point but the derivative is undefined. (What is the "slope" of the function at that point .....1  or -1 ??)

We can see this in one more way......the absolute value of a number "x" is defined as √(x2).....and taking the derivative of this gives us  .......   (1/2) (x2)-1/2 *(2x)  =  x / √(x2) =  x / lxl     ........and note that, this derivative is undefined at x = 0  !!!!

CPhill  Sep 3, 2014
#1
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Sorry it was a true or false question.

Guest Sep 3, 2014
#2
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AzizHusain  Sep 3, 2014
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Thank you.

Guest Sep 3, 2014
#4
+94118
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I'm just thinking.

If f(x) is continuous in an open region does that mean it could be a relation that is not  a function.

Like a circle or an s shape?

Relations that are not functions are not differentiable either.

Melody  Sep 3, 2014
#5
+92788
+5
Best Answer

As one of my calculus teachers would have said.....differentiability implies continuity, but continuity does not imply differentiability.....

To see why, look at the absolute value function at x = 0. Note that, it is continuous at that point but the derivative is undefined. (What is the "slope" of the function at that point .....1  or -1 ??)

We can see this in one more way......the absolute value of a number "x" is defined as √(x2).....and taking the derivative of this gives us  .......   (1/2) (x2)-1/2 *(2x)  =  x / √(x2) =  x / lxl     ........and note that, this derivative is undefined at x = 0  !!!!

CPhill  Sep 3, 2014
#6
+94118
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The absolute value of x is the one Aziz gave us as a counter example. (from the site he sent us to)

My query/consideration was a tad different.

Melody  Sep 3, 2014

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