When is a function not differentiable?

Thread starter Manni
Start date Oct 25, 2011
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Differentiable Function

In summary, the concept of differentiability in mathematics refers to the existence of a limit for the slope of the tangent line at a particular point in a function. A function can be differentiable at one point but not at another, depending on its continuity and the presence of sharp corners or holes. There are three types of discontinuities that make a function not differentiable: jump, removable, and essential. A function can also be differentiable but not continuous if it has a sharp corner or cusp. Additionally, not every continuous function is differentiable, as other conditions must be met for a function to be considered differentiable.

Oct 25, 2011

Manni

I'm curious about the conditions for when a function f(x) is not differentiable

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Oct 25, 2011

ArcanaNoir

At a point, or cusp.
At a non-removable discontinuity (such as a jump discontinuity).
On an asymptote.
Probably some other places too.

[Edit]
Also at a place where the slope is vertical.

Oct 26, 2011

daveb

How about any time it fails the definition of the derivative (i.e., the limit in the definition doesn't exist).

Related to When is a function not differentiable?

1. What is the definition of differentiability for a function?

According to the mathematical concept of differentiability, a function is considered differentiable at a point if the limit exists for the slope of the tangent line at that point. This means that the function is smooth and has a well-defined slope at that particular point.

2. Can a function be differentiable at one point but not at another?

Yes, a function can be differentiable at one point but not at another. This occurs when the function is not continuous or has a sharp corner at that point, making it impossible to define a unique tangent line and therefore not differentiable.

3. What are the types of discontinuities that make a function not differentiable?

The types of discontinuities that make a function not differentiable are jump discontinuities, removable discontinuities, and essential discontinuities. Jump discontinuities occur when there is a sudden jump in the function's value. Removable discontinuities occur when there is a hole in the graph of the function. Essential discontinuities occur when the limit of the function does not exist at a particular point.

4. How can a function be differentiable but not continuous?

A function can be differentiable but not continuous if it has a sharp corner or a cusp at a particular point. This means that although the slope of the tangent line exists, the function is not smooth and continuous at that point.

5. Is every continuous function differentiable?

No, not every continuous function is differentiable. A function must also satisfy additional conditions, such as being smooth and having a well-defined slope, to be considered differentiable. A function can be continuous but not have a well-defined slope, making it not differentiable.