Talk:Fundamental increment lemma

The assertion is FALSE because ...

"The lemma asserts that the existence of this derivative implies the existence of a function phi ..."

Let's take as an example f(x)= (x-1)^(1/3) + 1 for which f'(x) does not exist. This neither implies nor rejects the fact that a function phi(h) exists in any way whatsoever.

Q(x,h) = ( (x+h-1)^(1/3) - (x-1)^(1/3) )/h - ( (x-1)^(-2/3) )/3

It's not just that Wikipedia is a load of rubbish. It's that it is a dangerous load of rubbish.

"The lemma asserts that the existence of this derivative f(x) implies the existence of a function phi ..."

FALSE in its entirety. Whether derivative exists or not is immaterial to the function Q(x,h) or phi(h) as you incorrectly call it in this article because the function depends on both x and h.

The function Q(x,h) is made up of the terms containing a factor of h and may also contain x. However, Q(x,h) contains no terms without a factor of h. — Preceding unsigned comment added by RealMathematician (talk • contribs) 15:38, 9 September 2023 (UTC)[reply]

RealMathematician (talk) 14:28, 9 September 2023 (UTC)[reply]

What's wrong with the assertion "that the existence of this derivative implies the existence of a function phi such that" so-and-so? It's a rather simple fact. Actually, proven in the article.

Concerning your "example". It seems that you do not understand the logical significance of an "implication": The claim is "If P then Q". Considering a case with not-P is completely irrelevant concerning the validity of the original claim ("If P then Q"). 84.155.152.129 (talk) 15:03, 9 September 2023 (UTC)[reply]

So, this IP is a clown none other than Markus Klyver.

You can't reason with this type. He is an idiot.

The assertion is WRONG, you idiot! I am not here to teach you propositional logic. RealMathematician (talk) 15:36, 9 September 2023 (UTC)[reply]

John Gabriel usually calls himself "the greatest mathematician since Archimedes" but he doesn't understand how implications work. Clearly "P implies Q" isn't the same as "not P implies not Q". But John Gabriel doesnt understand this, unfortunately. This could explain why he fails to understand the limit definition or why he thinks "3 ⇔ 2+1" makes sense. Or why he thinks we can't say that 3≤4.

Of course, f'(a) must exist for the formula to make sense.

The lemma states that difference quotient = derivative at x=a + error, and this is a special case of the limit definition. 81.225.32.185 (talk) 15:37, 9 September 2023 (UTC)[reply]

"function Q(x,h) or phi(h) as you incorrectly call it in this article because the function depends on both x and h."

To say that a function depends on x and h or just h isn't a huge difference.

It entirely depends on what you consider fixed and what's not fixed. In the derivative definition, x is usually fixed and h is allowed to vary around the origin. 81.225.32.185 (talk) 15:41, 9 September 2023 (UTC)[reply]

Plagiarism by Talman.

The identity (f(x+h)-f(x))/h = f'(x) + Q(x,h) is my (John Gabriel) identity that is proved for all smooth functions in the Historic Geometric Theorem that was realised from the New Calculus.

The slope difference function is Q(x,h) and not Q(h).

www.academia.edu/62358358/My_historic_geometric_theorem_of_January_2020

From this identity, both the derivative and definite integral can be defined:

www.academia.edu/105576431/The_Holy_Grail_of_Calculus

Also, it seems strange that he is allowed to create this Wikipedia entry given that the only supporting source is a short article by Talman. I always thought that one could not create any entry without the source being a physical book or journal?

64.99.242.121 (talk) 22:31, 30 August 2023 (UTC)[reply]

You're a funny dude John Gabriel. No, Wikipedia won't delete it because of "plagiarism". But I agree that the lemma doesn't say much and is an immediate consequence of the limit definition. 81.225.32.185 (talk) 07:14, 5 September 2023 (UTC)[reply]

I don't see any humour but can understand that feeble minds are easily amused.

Wikipedia should delete it - period because it is NONSENSE.

You're wrong about it being a consequence of the limit definition. It's the other way around. RealMathematician (talk) 21:43, 5 September 2023 (UTC)[reply]

Explain why it's 'nonsense'. 81.225.32.185 (talk) 01:25, 7 September 2023 (UTC)[reply]

I have done so. See below. RealMathematician (talk) 10:05, 7 September 2023 (UTC)[reply]

I also want to add that this "lemma" is incorrect because the slope of a secant line DOES NOT depend on the existence of phi(h) or f'(a). In fact, f'(a) need not exist at all. Moreover, if a derivative exists, then the identity (f(x+h)-f(x))/h = f'(x) + Q(x,h) is provable for any smooth function f. See article above. — Preceding unsigned comment added by RealMathematician (talk • contribs) 11:46, 4 September 2023 (UTC)[reply]

Hint: The lemma does not state that "the slope of a secant line depends on the existence of phi(h) or f'(a). So your complaint is moot. In addition: "if a derivative exists [at the point a], then the identity (f(a+h)-f(a))/h = f'(a) + Q(a,h) is provable [...]". That's just what the lamma says! 79.206.207.80 (talk) 00:51, 8 September 2023 (UTC)[reply]

Moderator: The previous IP is a troll called Fritz Feldhase from sci.math. RealMathematician (talk) 11:42, 8 September 2023 (UTC)[reply]

It most certainly does! "The existence of the derivative" means tangent line and non-parallel secant line are present. The main stream derivative requires both the tangent and secant line. RealMathematician (talk) 12:51, 9 September 2023 (UTC)[reply]

Delete this article as it is incorrect.

I also want to add that this "lemma" is incorrect because the slope of a secant line DOES NOT depend on the existence of phi(h) or f'(a). In fact, f'(a) need not exist at all. Moreover, if a derivative exists, then the identity (f(x+h)-f(x))/h = f'(x) + Q(x,h) is provable for any smooth function f. See article above. RealMathematician (talk) 11:55, 4 September 2023 (UTC)[reply]

What's the incorrect part? 84.216.129.226 (talk) 16:41, 7 September 2023 (UTC)[reply]

Are you a moron or what? RealMathematician (talk) 16:57, 7 September 2023 (UTC)[reply]

Nope. Please be civil, as per Wikipedia guidelines, and explain why you think the lemma is stated incorrectly. 84.216.129.226 (talk) 20:49, 7 September 2023 (UTC)[reply]

Here: "...the slope of a secant line DOES NOT depend on the existence of phi(h) or f'(a). In fact, f'(a) need not exist at all. Moreover, if a derivative exists, then the identity (f(x+h)-f(x))/h = f'(x) + Q(x,h) is provable for any smooth function f.

www.academia.edu/62358358/My_historic_geometric_theorem_of_January_2020" RealMathematician (talk) 21:26, 7 September 2023 (UTC)[reply]

Hint: The lemma does not state that "the slope of a secant line depends on the existence of phi(h) or f'(a). So your complaint is moot. In addition: "if a derivative exists [at the point a], then the identity (f(a+h)-f(a))/h = f'(a) + Q(a,h) is provable [...]". That's just what the lamma says! 79.206.207.80 (talk) 00:43, 8 September 2023 (UTC)[reply]

Just like I thought. You're am imbecile with poor reading comprehension. 64.99.242.121 (talk) 06:32, 8 September 2023 (UTC)[reply]

This kind of language does not belong on Wikipedia. 84.216.157.17 (talk) 23:08, 8 September 2023 (UTC)[reply]

Moderator: This IP is a troll from sci.math called Fritz Feldhase. RealMathematician (talk) 11:41, 8 September 2023 (UTC)[reply]

I think you misunderstand the lemma. It does not say that. It says that if f'(a) if defined, we can write f'(a) as a secant + some error that goes to zero when h goes to zero. The error term is called phi, or Q.

I think you don't quite understand what the lemma actually says. 84.216.157.17 (talk) 23:08, 8 September 2023 (UTC)[reply]

Agree. 79.206.207.80 (talk) 02:05, 9 September 2023 (UTC)[reply]

I understand well. It is you who do not understand:

"The lemma asserts that the existence of this derivative implies the existence of a function phi(h) such that."

The assertion is false. phi(x,h) exists whether or not f'(a) exists. RealMathematician (talk) 12:46, 9 September 2023 (UTC)[reply]

Moreover, Q(x,h) is not an error term of any kind. It is a difference in slope between the tangent line and the function f. One can't even begin to talk about f'(a) unless it is given that f is smooth at every point in (x, x+h). RealMathematician (talk) 12:48, 9 September 2023 (UTC)[reply]

Sorry, between the tangent line and the non-parallel secant line anchored at x. RealMathematician (talk) 12:49, 9 September 2023 (UTC)[reply]