Is a this function differentiable?

[SprucerMoose]

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?

 

[Hyperbolful]

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

 

[MathWarrior]

End points on a closed interval would have no derivative.

 

[SprucerMoose]

Thanks a lot guys.

 

[Unrest]

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.

 

[AlephZero]

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.

 

[SprucerMoose]

So this is a semantic question

Do you have a problem with that?

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.