This is the Realm of Simply Beautiful Art

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2:53 AM

@user21820 Ah yes, I know the second one (the Devil's staircase). Is the 1st one the Weirstrauss function?

3:34 AM

@DavidReed It isn't, and I don't know if it has a name, but it sure looks more natural than the Weierstrass function. That's why I call it the broccoli fractal.

3 hours later…

Xander Henderson

6:31 AM

The first one is a "blancmange"

Blancmange curve

In mathematics, the blancmange curve is a fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name blancmange comes from its resemblance to a pudding of the same name. It is a special case of the more general de Rham curve. == Definition == The blancmange function is defined on the unit interval by b l a n c ...

it is clearly a french pudding, not broccoli

David Reed

@user21820 take note

user21820

6:52 AM

@DavidReed Aha thanks @XanderHenderson!

@XanderHenderson Do you have a picture of french pudding that looks like that? I could only find these.

7 hours later…

user21820

1:55 PM

@XanderHenderson: Do you have any idea why this post got downvoted. So far 3 people don't like it, for who knows what reason. Nobody is giving any concrete objection in the comments. And also:

@ParamanandSingh: See this post by Christian Blatter with a similar viewpoint although he's still using outdated terminology in "function of a hidden variable $t$ (time)". In fact one of the major reasons I devised my framework was to give a rigorous account of the common manipulations used in many places especially in mechanics such as when you have position, velocity, acceleration and differential equations relating them all. — user21820 Feb 17 '17 at 7:43

Xander Henderson

2:38 PM

@user21820 While I didn't downvote it, and I cannot really get into the head of whoever did, I kind of think that it runs a bit tangential to the original question

in that you introduce an entirely new theory of differentiation in order to give the symbols meaning in a larger context

since the OP didn't give any definitions, I suppose that is reasonable

user21820

@XanderHenderson Exactly, the asker didn't define anything, and used dx/dy just like that, and I felt that the only way to give a comprehensive answer that really encompasses the usual notions in physics where such derivatives are used was the generalized framework.

In particular, the asker asked for when one could make "dx/dy" make sense.

Xander Henderson

Sure, but there is clearly someone out there who disagrees

again, the question is so vague

typically, when students are introduced to differentiation, the terms $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $y'(x)$ are used interchangeably

with the Leibniz notation being a useful shorthand that lets you "get away" with symbolic manipulations that are not well justifiable at an introductory level (such as playing around with implicit differentiation)

because $\frac{\mathrm{d}y}{\mathrm{d}x} = y'(x)$ by definition, one would require $y$ to be a differentiable function of $x$ in order to deal with the question---it is implicit in the definition that is typically given to first year calc students

thus it is not wrong to require that $y$ be differentiable and nonzero at a point

(someone might have downvoted simply because you said that everyone else was wrong, I guess)

user21820

@XanderHenderson Yes that's of course true if the asker had that definition in mind. But then "dx/dy" would be meaningless; type-error.

Anyway I do know the standard calculus approach. Though I think it's a mistake to give up the intuitive natural of derivatives as limits of ratios on a parametrized curve. Otherwise so many things can't be done naturally. For example implicit differentiation in the sense of my post applies here:

A: Real life situation for an implicit function

A vast majority of scientific experiments can be considered to be collecting data that follows some implicit relation. What do I mean? In a typical scientific experiment, you wish to test or investigate an effect, and you have a number of parameters that you think can possibly determine some effe...

Xander Henderson

@user21820 You and I know that. The OP likey doesn't have the mathematical maturity to even know what that means.

user21820

I suppose so.

Xander Henderson

2:49 PM

At any rate, if I were you, I would not call the other answers wrong; I would note that they are assuming a very restrictive (and very silly) definition of the symbol $\frac{\mathrm{d}x}{\mathrm{d}y}$, then observe that this can (and really should) be defined better

user21820

Okay thanks for your feedback.

Xander Henderson

I would also fix all of your differentials---$dx$ looks silly :P

user21820

Haha okay. I'll be away for a while. See you!

Xander Henderson

laters

1 hour later…

user21820

3:58 PM

@XanderHenderson: I have improved the opening paragraph of my post. I don't really want to spend the time changing all the differentials though. In editing my post, I realized that I wasn't incorrect to say that the other answers were wrong, because my Example 1 actually furnishes a counter-example to the claim that we need differentiability in a neighbourhood of the point. We don't. So I've added a paragraph to that example explicitly stating that. =)

4 hours later…