Is a this function differentiable?

In summary, on a closed interval [a,b], a function that is continuous and differentiable for all x in R will have undefined derivatives at the endpoints a and b. This is due to the definition of the derivative at those points depending on function values outside of the interval. However, if the function is not defined outside the interval, then there can be a one-sided derivative at the endpoints.
  • #1
SprucerMoose
62
0
Hi all,

I was just wondering if a function that is continuous and differentiable for all xεR, but where the domain is restricted to closed interval [a,b], does the derivative exist at x=a or x=b?
 
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  • #2
It depends on your context.

According to Rudin (an analysis textbook author) there is a notion of a onesided derivative but in most cases the derivatives at the endpoints are underfined
 
  • #3
End points on a closed interval would have no derivative.
 
  • #4
Thanks a lot guys.
 
  • #5
MathWarrior said:
End points on a closed interval would have no derivative.

Is there a reason? Intuitively it seems like there should be a unique derivative, and you could evaluate it.
 
  • #6
If the function is differentiable for all x in R, then obviously it is differentiable at x = a and x = b.

But the definition of the derivative at x = a depends on function values at all points in an open interval containing a.

So this is a semantic question about what you mean by the word "restricted". First you said the function was differentiable everywhere, then you chose to ignore the fact that it was defined outside the interval [a,b] for some reason.

If the function is not defined outside the interval [a,b], then the most you can say is that there is a one-sided derivative at the end points of the interval.
 
  • #7
AlephZero said:
So this is a semantic question
Do you have a problem with that?

AlephZero said:
First you said the function was differentiable everywhere, then you chose to ignore the fact that it was defined outside the interval [a,b] for some reason.

I'm asking a technical question so i can gain a better understanding of how a derivative is defined. Your answer is concise and appreciated, but your condescending tone is not.
 

1. Is a function always differentiable?

No, not all functions are differentiable. A function must meet certain criteria in order to be considered differentiable, such as being continuous and having a defined derivative at every point in its domain.

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

A function is differentiable if it meets the criteria for differentiability, which includes being continuous and having a defined derivative at every point in its domain. You can also use the definition of differentiability, which states that a function is differentiable at a point if the left and right-hand limits of the function's derivative exist and are equal at that point.

3. What does it mean for a function to be non-differentiable?

A non-differentiable function is one that does not meet the criteria for differentiability, such as being discontinuous or having an undefined derivative at one or more points in its domain. This means that the function cannot be differentiated at those points.

4. Can a function be differentiable at some points and non-differentiable at others?

Yes, a function can be both differentiable and non-differentiable at different points in its domain. For example, a function may be differentiable everywhere except for at a point where it has a sharp corner or a vertical tangent.

5. Why is differentiability important in mathematics?

Differentiability is important in mathematics because it allows us to calculate the rate of change of a function at a specific point. This is useful in many applications, such as finding maximum and minimum values, optimization problems, and graphing functions. It also allows us to approximate the behavior of a function near a point using its derivative.

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