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Analysis help  Continuous function that is differentiable at all points except c 
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#1
Jan911, 08:01 PM

P: 69

1. The problem statement, all variables and given/known data
Let I be an interval, and f: I > R be a continuous function that is known to be differentiable on I except at c. Assume that f ' : I \ {c} > R admits a continuous continuation to c (lim x > c f ' exists). Show that f is in fact also differentiable at x and f ' (c) = lim x>c f '. 3. The attempt at a solution This seems like a very easy question to me, but for some reason its stumping me, maybe because of the way my prof worded it, but im just a little confused. I know i need to use the mean value theorem, but im still stuck. Please help. 


#2
Jan911, 09:29 PM

P: 850

It seems that the fact that (lim x > c f ' exists) means f derievative is bounded on I is important.
If I am thinking correctly I think f ' is uniformly continous since it has a continous extension on I. 


#3
Jan911, 09:41 PM

P: 69




#4
Jan911, 09:46 PM

P: 850

Analysis help  Continuous function that is differentiable at all points except c
Also if f' actually turns out to be uniformly continuous then the problem is trivial since f ' would be continous and which implies f ' (c) = lim x>c f '. 


#5
Jan1011, 08:21 AM

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Thanks
PF Gold
P: 39,310

While the derivative of a function is not necessarily continuous, it does satisfy the "intermediate value property": if f'(a)= c and g'(b)= d, then, for any e between c and d, there exist x between a and b such that f'(x)= e.
In particular, that means that f is differentiable at x= c if and only if [math]\lim_{x\to c^}f'(x)=\lim_{x\to c^+} f'(x)[/itex] and f'(c) is equal to that mutual value. 


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