How Does Differentiability Define a Function's Behavior?

[HIGHLYTOXIC]

What does the diffrentiability of a given function at some point suggest? What more can we find out about a function if we are given the diffrentiability or non-differentiability at some point?

I have some idea on it like the slopes of the tangents to the curve differ, there may be sudden dips & sharp turns..But I can't apply them in questions dealing with the concept of diffrentiability..

Can anyone help with this concept?

 

[matt grime]

Differentialbility at a point is purely a local datum, it tells you nothing about the global behaviour of the function.

 

[Ethereal]

Doesn't the differentiability of the function at a point tell you that it is continuous at that point?

 

[Galileo]

More surprisingly. For sufficiently 'nice' functions, the derivatives of all orders at a given point determines the entire function in its domain.

 

[HallsofIvy]

More surprisingly. For sufficiently 'nice' functions, the derivatives of all orders at a given point determines the entire function in its domain.

"sufficiently 'nice'" being defined as a function such that the derivatives of all orders at a given point determines the entire function!
(i.e. "analytic")

 

[Hurkyl]

It's a fair description, though, because most of the functions most people could imagine are analytic 'most everywhere.

 

[matt grime]

An example of a family of functions, defined on the strictly positive reals, that shows just how local differentiability is:

Let k be any real number, and let f(x,k) be zero if x is irrational and k/n if x is rational and x=m/n where m and n are coprime, then f(x,k) is almost everywhere differentiable with derivative 0, and as long as k is not zero, is not differentiable at every rational number in the domain.Thus each of (the infinitely many) fs has the same domain, the same subset of the domain on which it is differentiable,with the same derivative, and they are all distinct.


EDIT actually I'm having second thoughts about this function, but it the above is true with the word continuous inserted for the word differentiable, and I'm too tired to think about it.