Showing f is Differentiable at c: A Challenge

In summary, if a continuous function f on an interval I is known to be differentiable at all points except c, where its derivative f' has a continuous extension to c, then f is actually differentiable at c and f'(c) is equal to that extension.
  • #1

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.
 
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  • #2
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.
 
  • #3
╔(σ_σ)╝ said:
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.
 
  • #4
cooljosh2k2 said:
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 '.
 
  • #5
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 \(\displaystyle \lim_{x\to c^-}f'(x)=\lim_{x\to c^+} f'(x)[/itex] and f'(c) is equal to that mutual value.\)
 

1. What is the definition of differentiability at a point?

The definition of differentiability at a point is when the limit of the difference quotient (or the slope of the tangent line) exists at that point. In other words, the function is smooth and has a well-defined slope at that point.

2. How do you prove that a function is differentiable at a point?

To prove that a function is differentiable at a point, you can use the definition of differentiability and show that the limit of the difference quotient exists at that point. Additionally, you can also use the differentiability rules, such as the sum, product, and chain rule, to show that the function is differentiable at that point.

3. What is the significance of differentiability in calculus?

Differentiability is significant in calculus because it allows us to find the slope of a tangent line at a specific point on a graph. This information is essential for many applications in science, engineering, and economics, such as optimization and rate of change.

4. Can a function be continuous but not differentiable at a point?

Yes, a function can be continuous but not differentiable at a point. This means that the function is smooth and has no breaks or holes at that point, but the slope of the tangent line does not exist. An example of this is the absolute value function at x=0.

5. Is it possible for a function to be differentiable at a point but not continuous?

No, it is not possible for a function to be differentiable at a point but not continuous. This is because a function must be continuous in order to be differentiable. If a function is not continuous at a point, then it cannot have a well-defined tangent line and therefore cannot be differentiable at that point.

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