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- Thread starter Treadstone 71
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Hurkyl

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Have you seen the Koch Snowflake, or any other fractal curve?

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The Weierstrass function is a related pathology. Weierstrass was one of the mathematicians (including Dedekind, Cantor, Kronecker, and so forth) that heralded the 2nd age of rigor, which put on firm ground many concepts that were previously nebulous in definition and application. Many pathologies were created at this time.

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[tex]\forall x\in\mathh{R}\mbox{ let } f_{0}(x)[/tex] be the distance from

For each [tex]n\in\mathbb{N}[/tex], define [tex]f_{n}(x)=\frac{f_{0}(12^{n}x)}{2^{n}}[/tex], then [tex]f(x)=\sum_{n=0}^{\infty} f_{n}(x)[/tex] is such a function. (continuity is from uniform convergence)

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quasar987

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This might not be the most on-topic post ever but lookit it's pretty funny:hypermorphism said:

The Weierstrass function is a related pathology. Weierstrass was one of the mathematicians (including Dedekind, Cantor, Kronecker, and so forth) that heralded the 2nd age of rigor, which put on firm ground many concepts that were previously nebulous in definition and application. Many pathologies were created at this time.

:rofl: :rofl: :rofl: :rofl:My applied analysis textbook cover said:Mr.Cauchy anounces that, in order to conform to the will of the Council, he will not anymore take care of providing, as he has done up until now, perfectly rigourous proofs of his statements.

Council of instruction of l'École Polytechnique,

november 24 1825

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Can you even take derivative if it is not continuous?

Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.

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NateTG

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There are functions that have the same derivative on both sides of discontinuities (like the greatest integer function), but the derivative at the discontinuity is undefined.leon1127 said:

Can you even take derivative if it is not continuous?

Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.

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...continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

The above quote is from http://mathworld.wolfram.com/Derivative.html

So in what "sense" is that [the " "more" " part]? I asked my Real analysis prof, he said in the sense of catagory. But I wonder, could it be true in the sense of cardinality? Specifically, is the cardinality of the set of all functions which are integrable greater than that of differentiable functions?

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Continuity is a neccessary, but not a sufficient condition for differentability. So no, you cannot differentiate a non-continuous function.leon1127 said:

Can you even take derivative if it is not continuous?

Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.

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So it means that suck function doesnt exist?benorin said:Continuity is a neccessary, but not a sufficient condition for differentability. So no, you cannot differentiate a non-continuous function.

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What's a suck function?

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I think he meant "such" function.

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lmao :rofl:

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If I ever create an original pathology, its name shall be suck. :rofl:

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Well it's technically not a function, but how about 0y+3x=6Treadstone 71 said:

it's a straight line, with an infinite value both plus and minus for every x different than 2.

Nah...that's not what you're looking for...what about f(x)=0?

I don't know...i tried.

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HallsofIvy

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Just about every calculus book has a proof that a function is differentiable at c only if it is continuous at c. There is no such function.leon1127 said:

Can you even take derivative if it is not continuous?

Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.

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