What is, and what is not integrable/differentiable?

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In summary, differentiation is not defined for all functions. There exist functions that are continuous everywhere but differentiable nowhere, as well as functions that are differentiable everywhere except at a certain point. A set of measure zero refers to a set of points that has no area, and a null set refers to an empty set.
  • #1
dimensionless
462
1
I have a problem that asks me to show that a function is differentiable. Aren't all functions differentiable

:confused:
 
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  • #2
No, they are not.
How is differentiation defined?
 
  • #3
In fact, there exist functions that are continuous everywhere and differentiable nowhere.

Regards,
George
 
  • #4
And, there exist nowhere continuous functions as well. :smile:
 
  • #5
dimensionless said:
I have a problem that asks me to show that a function is differentiable. Aren't all functions differentiable

:confused:

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

[tex]f(x) = x\sin\left(\frac{1}{x}\right) \;\; x \neq 0 \;\; f(0)=0[/tex]

However, [itex]f(x) = |x|[/itex] is differentiatble anywhere except at [itex]x = 0[/itex]

~H
 
  • #7
Hootenanny said:
FThis is continuous and is not differentiatble anywhere

[tex]f(x) = x\sin\left(\frac{1}{x}\right) \;\; x \neq 0 \;\; f(0)=0[/tex]

looks pretty differentiable everywhere but 0 to me since it is the composite and product of functions that are all differentiable away from 0.
 
  • #8
benorin said:
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.
 
  • #9
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...
 
  • #10
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.
 
  • #11
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 :blushing: .

~H
 
  • #12
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.
 
  • #13
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
 
  • #14
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!
 
  • #15
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).
 
  • #16
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.
 
  • #17
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
 
  • #18
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).
 
  • #19
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
 
  • #20
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!
 
  • #21
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.
 
  • #22
A very simple function that is continuous everywhere but not differentiable is f(x)= |x|.
 

1. What is the difference between integrable and differentiable functions?

Integrable and differentiable functions are two different types of mathematical functions. An integrable function is one that can be integrated, or have an antiderivative, while a differentiable function is one that can be differentiated, or have a derivative. In simpler terms, an integrable function can be thought of as a function that can be "reverse-engineered" to find its original function, while a differentiable function can be thought of as a function that has a slope at every point along its graph.

2. Can a function be both integrable and differentiable?

Yes, a function can be both integrable and differentiable. In fact, many common mathematical functions, such as polynomials and exponential functions, are both integrable and differentiable. This means that they have both an antiderivative and a derivative.

3. What types of functions are not integrable or differentiable?

There are certain types of functions that are not integrable or differentiable. These include functions that have sharp corners or discontinuities, such as absolute value functions, as well as functions that are undefined at certain points, such as the inverse tangent function. These types of functions cannot be integrated or differentiated using traditional methods.

4. What is the importance of integrable and differentiable functions in mathematics?

Integrable and differentiable functions are important in mathematics because they allow us to model and solve various real-world problems. For example, integrable functions can be used to find the area under a curve, while differentiable functions can be used to determine rates of change. These functions also play a crucial role in fields such as physics, engineering, and economics.

5. How can you determine if a function is integrable or differentiable?

Determining if a function is integrable or differentiable can be done using various techniques. For integrable functions, you can check if the function has a continuous derivative or if it satisfies certain conditions, such as the Fundamental Theorem of Calculus. For differentiable functions, you can use the limit definition of the derivative or check for the existence of a tangent line at each point on the graph. In some cases, it may also be necessary to use more advanced mathematical techniques to determine if a function is integrable or differentiable.

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