Confusion while studying Vector Calculus from Hubbard & Hubbard
Monday 22 September 2025 at 16:00
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Edited by Zafar Bakhromov, Monday 22 September 2025 at 16:02
On page 93 of John & Barbara Hubbard's Vector Calculus, Linear Algebra and Differential Forms, I noticed something unusual. The authors state that functional limits behave well under composition (Theorem 1.5.22).
More formally, let be subsets, and and be mappings, so that is defined . If is a point in and
and
both exist, then exists, and
At first, this seemed false, because in order for it to be true the function needs to be continuous. But no such assumption is made here. I was scratching my head the whole afternoon trying to figure this out, and after a long time of googling, I found the following result in wikipedia:
It seems like whether whenever is near is important. So I looked back on the definition of function limits in the book, and I found the key fact that was confusing me: the authors did not require whenever is near ! Appearently this definition of functional limits is common in France, while the usual definition is common in the United States.
As an example, consider the functions
and
Then, we get the following table of facts:
French Definition
US Definition
exists?
Yes
Yes
exists?
No
Yes
exists?
No
No
Is Theorem 1.5.22 True?
Yes (vacuously: the hypotheses aren't met)
No
It is straightforward to show that does not exist under the French definition, because for any we can always find an s.t. (Here, is either equal to 0 or 1). Hence, the theorem is true.
This little detour reminded me that even definitions we take for granted can vary by culture, and those small differences can have big consequences. For me, it was a reminder that part of learning mathematics is also learning to pay very close attention to the precise conventions being used.
Confusion while studying Vector Calculus from Hubbard & Hubbard
On page 93 of John & Barbara Hubbard's Vector Calculus, Linear Algebra and Differential Forms, I noticed something unusual. The authors state that functional limits behave well under composition (Theorem 1.5.22).
More formally, let be subsets, and and be mappings, so that is defined . If is a point in and
and
both exist, then exists, and
At first, this seemed false, because in order for it to be true the function needs to be continuous. But no such assumption is made here. I was scratching my head the whole afternoon trying to figure this out, and after a long time of googling, I found the following result in wikipedia:
It seems like whether whenever is near is important. So I looked back on the definition of function limits in the book, and I found the key fact that was confusing me: the authors did not require whenever is near ! Appearently this definition of functional limits is common in France, while the usual definition is common in the United States.
As an example, consider the functions
and
Then, we get the following table of facts:
It is straightforward to show that does not exist under the French definition, because for any we can always find an s.t. (Here, is either equal to 0 or 1). Hence, the theorem is true.
This little detour reminded me that even definitions we take for granted can vary by culture, and those small differences can have big consequences. For me, it was a reminder that part of learning mathematics is also learning to pay very close attention to the precise conventions being used.