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Analysis help - Continuous function that is differentiable at all points except c

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cooljosh2k2
#1
Jan9-11, 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.
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╔(σ_σ)╝
#2
Jan9-11, 09:29 PM
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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.
cooljosh2k2
#3
Jan9-11, 09:41 PM
P: 69
Quote Quote by ╔(σ_σ)╝ View Post
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.
How does the f ' being uniformly continuous help me at reaching my answer? If the Interval is [a,b], then the f ' is continuous on the open intervals (a,c) and (c,b), how could i show that while f' may not be continuous at in the interval at c, a derivative still exists.

╔(σ_σ)╝
#4
Jan9-11, 09:46 PM
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P: 850
Analysis help - Continuous function that is differentiable at all points except c

Quote Quote by cooljosh2k2 View Post
How does the f ' being uniformly continuous help me at reaching my answer? If the Interval is [a,b], then the f ' is continuous on the open intervals (a,c) and (c,b), how could i show that the f ' is continuous from (a,b) and therefore a derivative exists at c.
If we assume (lim x -> c f ' exists) then f ' has to be continuous at c since the left and right limits have to be equal. Once f ' is continuous on (a,b), f ' (c) = lim x->c f ' is simply a consequence of continuity.


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 '.
HallsofIvy
#5
Jan10-11, 08:21 AM
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PF Gold
P: 39,345
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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