# 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

Homework Helper
Gold Member
Dearly Missed
No, they are not.
How is differentiation defined?

#### George Jones

Staff Emeritus
Gold Member
In fact, there exist functions that are continuous everywhere and differentiable nowhere.

Regards,
George

#### arildno

Homework Helper
Gold Member
Dearly Missed
And, there exist nowhere continuous functions as well. #### Hootenanny

Staff Emeritus
Gold Member
dimensionless said:
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

Homework Helper

#### matt grime

Homework Helper
Hootenanny said:
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.

Homework Helper

#### Hootenanny

Staff Emeritus
Gold Member
matt grime said:
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

Homework Helper
Hootenanny said:
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

Staff Emeritus
Gold Member
matt grime said:
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

Homework Helper
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

Staff Emeritus
Gold Member
matt grime said:
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

Homework Helper
Gold Member
matt grime said:
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

Homework Helper
nrqed said:
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

Homework Helper
Gold Member
Dearly Missed
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

Staff Emeritus
Gold Member
Curious3141 said:
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

Homework Helper
George Jones said:
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

Staff Emeritus
Gold Member
Curious3141 said:
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

Homework Helper
Gold Member
arildno said:
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!

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving