Basic differentiability question

In summary, the question is whether a continuous function with a limit of its derivative at 0 must also have a derivative at 0. The hint is to consider the difference quotient and the Mean Value Theorem.
  • #1
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Homework Statement



Let f: R -> R be a continuous function such that f '(x) exists for all x =/= 0 . Say also that the limit of f '(x) as x goes to 0 exists and is equal to L. Must f '(0) exist as well? Prove or disprove.

The Attempt at a Solution


I can't come up with a proof or counterexample. It seems like it must be true but I've learned to not completely trust my intuition when it comes to these things (pathological counterexamples come to mind). Can anyone give me a hint on whether or not this is true or how to go about proving/disproving it?
 
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  • #2
Think about the difference quotient (f(h)-f(0))/h as h->0. If that does not converge to 0 then the there is a sequence h_i->0 such that the difference quotient is bounded away from 0. Now think about the Mean Value Theorem. Remember that the premises of the MVT don't require differentiability at the endpoints.
 
  • #3
Got it. thanks a lot, that was perfect.
 

1. What is basic differentiability?

Basic differentiability refers to the property of a mathematical function to have a well-defined derivative at a certain point. This means that the function is smooth and continuous at that specific point, and its rate of change can be calculated.

2. How is differentiability different from continuity?

Continuity refers to the property of a function to have no abrupt changes or breaks. A function can be continuous but not differentiable, meaning that it is smooth and continuous but does not have a well-defined derivative at a certain point. Differentiability implies continuity, but not vice versa.

3. What is the importance of differentiability in calculus?

Differentiability is essential in calculus because it allows us to calculate the rate of change of a function at a specific point. This is necessary for many applications, such as optimization, finding maximum and minimum values, and solving differential equations.

4. How can I determine if a function is differentiable?

A function is differentiable at a point if its derivative exists at that point. To determine if a function is differentiable, you can use the definition of the derivative or check for the differentiability conditions, such as continuity and smoothness, at that point.

5. Can a function be differentiable but not continuous?

No, a function cannot be differentiable at a point if it is not continuous at that point. This is because differentiability implies continuity, and if a function has a well-defined derivative, it must also be smooth and continuous at that point.

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