How Does Differentiability Define a Function's Behavior?

  • Thread starter HIGHLYTOXIC
  • Start date
  • Tags
    Function
In summary, the differentiability of a function at a point suggests that it is continuous at that point, and for sufficiently "nice" functions, the derivatives of all orders at a given point determines the entire function in its domain.
  • #1
HIGHLYTOXIC
47
0
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?
 
Physics news on Phys.org
  • #2
Differentialbility at a point is purely a local datum, it tells you nothing about the global behaviour of the function.
 
  • #3
Doesn't the differentiability of the function at a point tell you that it is continuous at that point?
 
  • #4
More surprisingly. For sufficiently 'nice' functions, the derivatives of all orders at a given point determines the entire function in its domain.
 
  • #5
Galileo said:
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")
 
  • #6
It's a fair description, though, because most of the functions most people could imagine are analytic 'most everywhere.
 
  • #7
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.
 
Last edited:

FAQ: How Does Differentiability Define a Function's Behavior?

What is the definition of differentiability of a function?

The differentiability of a function is a mathematical concept that describes how smoothly a function changes at a given point. A function is said to be differentiable at a point if it has a well-defined tangent line at that point, meaning that the function is continuous and has a defined derivative at that point.

How is differentiability related to continuity?

Differentiability is closely related to continuity. A function must be continuous at a point in order to be differentiable at that point. However, a function can be continuous at a point without being differentiable at that point. This means that differentiability is a stronger condition than continuity.

What is the difference between a differentiable function and a continuously differentiable function?

A differentiable function is one that has a well-defined derivative at a given point. A continuously differentiable function is one that has a derivative that is also continuous at that point. This means that a continuously differentiable function is both differentiable and continuous at a given point.

Can a function be differentiable at some points but not others?

Yes, a function can be differentiable at some points and not others. For example, a function can be differentiable at all points except for a single point where it has a sharp corner or a discontinuity. In this case, the function is still considered differentiable, but not at that specific point.

How can I determine if a function is differentiable?

To determine if a function is differentiable at a given point, you can use the definition of differentiability. This involves checking if the function is continuous at that point and if it has a well-defined derivative at that point. You can also use the differentiability rules and techniques, such as the chain rule and the product rule, to determine if a function is differentiable at a given point.

Similar threads

B Problem with the concept of differentiation
2
Replies
46
Views
4K
B Why does the Limit Process give an Exact Slope?
2
Replies
53
Views
2K
I Partial derivatives of the function f(x,y)
Replies
6
Views
2K
B Can the continuity of functions be defined in the field of rational numbers?
Replies
5
Views
2K
B Confusion on Implicit Differentiation
Replies
16
Views
2K
I Approximation of a function of two variables
Replies
3
Views
1K
B Tangents to a curve as functions of different variables
Replies
5
Views
1K
B Functions which relate to calculus: Questions about Notation
2
Replies
36
Views
5K
MHB Calculus & Diff. Eqn: Beginner Qs on Function, Derivative & Gradient
Replies
14
Views
3K
I How Does Gaussian-Legendre Quadrature Approximate Non-Polynomial Functions?
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top