What's the definition of a point being differentiable?

In summary, we are trying to prove that for a function f that is continuous on [a,b] and differentiable at all points of (a,b) except possibly at c, where lim(x->c)f'(x) exists and is equal to k, f is also differentiable at c and f'(c)=k. This can be shown using the intermediate value property of f' and the mean value theorem on (x0, c) and (c, x1) to demonstrate that the left and right limits of the difference quotient exist and are equal, thus proving the existence and equality of f'(c).
  • #1
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"Suppose f is continuous on [a,b] and c in (a,b). Suppose f is differentiable at all points of (a,b) except possibly at c. Assume further that lim(x->c)f'(x) exists and is equal to k. Prove that f is differentiable at c and f'(c)=k"

Since the lim f'(x) as x->c exists, f'(c) either equals k, exists but doesn't equal to k, or undefined. I showed that if it is defined, it must equal to k by using the intermediate value property of f'. But I can't show that f'(c) has to be defined. I tried contradiction, saying if f'(c) is undefined, but I can't run into a contradiction.
 
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  • #2
What's the definition of a point being differentiable?
 
  • #3
lim(x->c) (f(x)-f(c))/(x-c) = f'(c), provided the limit exists.
 
  • #4
Try using the mean value theorem on (x0, c) and (c, x1) to show that the left and right limits of the difference quotient exist and are the same.
 
  • #5
I'm not sure what you mean by difference quotient. The mean value theorem assumes the existence of f'(c).
 

What is the definition of a point being differentiable?

The definition of a point being differentiable is that it is a mathematical concept that describes the smoothness of a function at a specific point. A point is said to be differentiable if the function has a well-defined tangent line at that point.

What is the difference between differentiability and continuity?

Differentiability and continuity are related concepts, but they are not the same. A function is continuous if it is defined at a point and its limit is equal to the value of the function at that point. A function is differentiable if it has a well-defined tangent line at a specific point. In other words, continuity means that the function is "connected" without any breaks, while differentiability means that the function is "smooth" at a specific point.

How is differentiability related to the slope of a function?

The slope of a function is the steepness of the curve at a specific point. A function is differentiable at a point if and only if the slope of the function at that point is well-defined. This means that the function has a tangent line at that point, which represents the slope of the function at that point.

What are the necessary conditions for a function to be differentiable?

For a function to be differentiable at a point, it must first be continuous at that point. Additionally, the function must also have a well-defined slope at that point, meaning that the limit of the slope as the distance between points approaches zero must exist. If these two conditions are met, then the function is differentiable at that point.

Can a function be differentiable at one point but not at another?

Yes, a function can be differentiable at one point but not at another. This is because differentiability depends on the existence of a well-defined tangent line at a specific point, and this may vary from point to point on a function. A function can be differentiable at one point and not at another if the conditions for differentiability are not met at the other point.

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