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May 20th, 2012, 02:31 AM   #1
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differentiability

1) What happens if a function has a jump discontinuity but the left and right side of the derivative are equal?
2) What happens if a function has no critical points or the derivative has complex roots for e.g. f(x)=x^4+x^2+x+1 => f'(x)=4x^3+2x+1?
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May 20th, 2012, 05:56 AM   #2
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Re: differentiability

Quote:
Originally Posted by alexmath
1) What happens if a function has a jump discontinuity but the left and right side of the derivative are equal?
Example: a step function
Quote:
Originally Posted by alexmath
2) What happens if a function has no critical points or the derivative has complex roots for e.g. f(x)=x^4+x^2+x+1 => f'(x)=4x^3+2x+1?
Each of these is the sum of where
on
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May 20th, 2012, 12:10 PM   #3
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Re: differentiability

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Originally Posted by greg1313
1) A function is not differentiable at a discontinuity - you must take the limit from both sides of the point, i.e. the left and right hand limits must exist and be equal for the derivative to exist at that point.
That's not a jump discontinuity.
Consider
on bothsides of 0. The and
There is a jump of 2 units from the left hand side to the right hand side.
Since does not exist, it's discontinuous apart from the jump.

exists in and
Quote:
Originally Posted by greg1313
2) If a function has no critical points the function is either increasing everywhere (positive derivative) or decreasing everywhere (negative derivative).
A constant function has no critical point. It's differetiable; that is, . A constant function is neither increasing nor decreasing.
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May 20th, 2012, 01:23 PM   #4
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Re: differentiability

I've deleted the post. Thanks for the correction.
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