Showing f is Differentiable at c: A Challenge

[cooljosh2k2]

Homework Statement

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 '.

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 I am just a little confused. I know i need to use the mean value theorem, but I am still stuck. Please help.

 

[╔(σ_σ)╝]

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 continuous since it has a continuous extension on I.

 

[cooljosh2k2]

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 continuous since it has a continuous 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)╝]

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 continuous and which implies f ' (c) = lim x->c f '.

 

[HallsofIvy]

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.