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 May 20th, 2012, 02:31 AM #1 Senior Member   Joined: Apr 2012 Posts: 135 Thanks: 1 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?
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 $x^n$ where $n=0,1,2,\cdots$
$\frac{d}{dx}x^n=nx^{n-1}$ on $(-\infty, \infty)$

May 20th, 2012, 12:10 PM   #3
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Re: differentiability

Quote:
 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 $f(x)=\left \{\begin{array}{cc}
1& x>0\\
-1& x<0 \end{array}\right .$

$\frac{dy}{dx}=0$ on bothsides of 0. The $\displaystyle \lim_{x+\rightarrow 0}=-1$ and $\lim_{x-\rightarrow 0}=1$
There is a jump of 2 units from the left hand side to the right hand side.
Since $f(0)$ does not exist, it's discontinuous apart from the jump.

$\frac{dy}{dx}$ exists in $(0, \infty)$ and $(-\infty,0)$
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 $f(x)=cx^{0}=c$ has no critical point. It's differetiable; that is, $\frac{dy}{dx}=0$. A constant function is neither increasing nor decreasing.

 May 20th, 2012, 01:23 PM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond Re: differentiability I've deleted the post. Thanks for the correction.

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