What is, and what is not integrable/differentiable?

[dimensionless]

I have a problem that asks me to show that a function is differentiable. Aren't all functions differentiable

 

[arildno]

No, they are not.
How is differentiation defined?

 

[George Jones]

In fact, there exist functions that are continuous everywhere and differentiable nowhere.

Regards,
George

 

[arildno]

And, there exist nowhere continuous functions as well.

 

[Hootenanny]

I have a problem that asks me to show that a function is differentiable. Aren't all functions differentiable

Further to George's comment;

In fact, there exist functions that are continuous everywhere and differentiable nowhere.

This is continuous and is not differentiatble anywhere

$$f(x) = x\sin\left(\frac{1}{x}\right) \;\; x \neq 0 \;\; f(0)=0$$

However, $f(x) = |x|$ is differentiatble anywhere except at $x = 0$

~H

 

[benorin]

The Weierstrass function and the Blancmange Function are examples of everywhere continuous yet nowhere differentiable functions

 

[matt grime]

FThis is continuous and is not differentiatble anywhere

$$f(x) = x\sin\left(\frac{1}{x}\right) \;\; x \neq 0 \;\; f(0)=0$$

looks pretty differentiable everywhere but 0 to me since it is the composite and product of functions that are all differentiable away from 0.

 

[matt grime]

The Weierstrass function and the Blancmange Function are examples of everywhere continuous yet nowhere differentiable functions

the first of those is differentiable on a set of measure zero as the link you posted informs you.

 

[Hootenanny]

looks pretty differentiable everywhere but 0 to me since it is the composite and product of functions that are all differentiable away from 0.

Runs off to do some checking...

 

[matt grime]

Runs off to do some checking...

Why? 1/x is differentiable everywhere but 0, sin is differentiable at all points x is differentiable at all points hence xsin(1/x) is differentiable at all points except where x=0.

 

[Hootenanny]

Why? 1/x is differentiable everywhere but 0, sin is differentiable at all points x is differentiable at all points hence xsin(1/x) is differentiable at all points except where x=0.

The reason I was checking was because I have this function written down as a non-differentiable function given as an example by my tutor, but it appears either she is wrong or I have copied down wrong (more likely). I've never thought to check it, till now .

~H

 

[matt grime]

Oh, the function is certainly not differentiable, but that is strictly different from nowhere differentiable, or not differentiable anywhere. A function is differentiable if it is differentiable at every point of its domain. So it only takes one point where it is not differentiable for the function to be 'not differentiable', yet it is differentiable everywhere except that one point.

 

[Hootenanny]

Oh, the function is certainly not differentiable, but that is strictly different from nowhere differentiable, or not differentiable anywhere. A function is differentiable if it is differentiable at every point of its domain. So it only takes one point where it is not differentiable for the function to be 'not differentiable', yet it is differentiable everywhere except that one point.

Make sense, thank's matt.

~H

 

[nrqed]

the first of those is differentiable on a set of measure zero as the link you posted informs you.

would it be possible to explain what one means by "differentiable on a set of measure zero"? Or to point to a website explaining this?
Thanks!

 

[Curious3141]

would it be possible to explain what one means by "differentiable on a set of measure zero"? Or to point to a website explaining this?
Thanks!

This says something about it. http://en.wikipedia.org/wiki/Measure_zero

My understanding (not great) from that is that a set of measure zero is necessarily a null set (which is important here, there are no points at which the function is differentiable). The converse is not necessarily true (which doesn't seem important here).

 

[arildno]

Roughly spoken, the measure of a set says how big it is.
In the plane, we might say that the measure of a set of points is the area of the region consisting of those points.

Consider now a line lying in the plane.
What is the area of that line?
Clearly, a line should have zero area, and this corresponds to saying that the set of points constituting the line is a set of measure zero.

 

[George Jones]

My understanding (not great) from that is that a set of measure zero is necessarily a null set (which is important here, there are no points at which the function is differentiable). The converse is not necessarily true (which doesn't seem important here).

I think you've got it the wrong way round.

If A is the null set, then necessarily A has measure zero, but, if A has measure zero, then A is not necessarily the null set. For example, if A is the set of all rational number, then, as a subset of the standard measure space of real numbers, A has measure zero.

Regards,
George

 

[Curious3141]

I think you've got it the wrong way round.

If A is the null set, then necessarily A has measure zero, but, if A has measure zero, then A is not necessarily the null set. For example, if A is the set of all rational number, then, as a subset of the standard measure space of real numbers, A has measure zero.

Regards,
George

Not according to the Wiki article I linked. Is Wiki wrong? (I honestly don't know, I was just lifting from there).

 

[George Jones]

Not according to the Wiki article I linked. Is Wiki wrong? (I honestly don't know, I was just lifting from there).

Okay, in this context, Wiki defines a null set to be a set of measure zero. Consequently, in my example, the set of all rational numbers is a null set.

But from

there are no points at which the function is differentiable

I took it that you meant the empty set, which is also sometimes called the null set.

Measure zero sets are not necessarily empty.

Regards,
George

 

[nrqed]

Roughly spoken, the measure of a set says how big it is.
In the plane, we might say that the measure of a set of points is the area of the region consisting of those points.

Consider now a line lying in the plane.
What is the area of that line?
Clearly, a line should have zero area, and this corresponds to saying that the set of points constituting the line is a set of measure zero.

Thanks, I had some idea of measure in general but I was wondering about "*differentiable* on a s et of measure zero". This means that the functions *is* differentiable on a (possibly infinite) number of points but that those points form a set of measure zero. Ok, but I guess I wonder how that looks like or how the proof is done.

Thanks for the help!

 

[matt grime]

You work out where it is differentiable, and show it has measure zero... For instance any countable set (such as the rationals) has measure zero.

 

[HallsofIvy]

A very simple function that is continuous everywhere but not differentiable is f(x)= |x|.