- #1
dimensionless
- 462
- 1
I have a problem that asks me to show that a function is differentiable. Aren't all functions differentiable
dimensionless said:I have a problem that asks me to show that a function is differentiable. Aren't all functions differentiable
In fact, there exist functions that are continuous everywhere and differentiable nowhere.
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]
benorin said:The Weierstrass function and the Blancmange Function are examples of everywhere continuous yet nowhere differentiable functions
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.
Hootenanny said:Runs off to do some checking...
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.
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.
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?matt grime said:the first of those is differentiable on a set of measure zero as the link you posted informs you.
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!
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).
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
Curious3141 said:Not according to the Wiki article I linked. Is Wiki wrong? (I honestly don't know, I was just lifting from there).
there are no points at which the function is differentiable
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.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.
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.
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.
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.
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.
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.