Identifying the name of a theorem

[nuuskur]

I can't find sufficient material for this theorem in my own languages so I have to resort to English, however, googling "Heine criterion" yields nothing significant so I think, once again I just can't get its English name right.

Limit of a function: Heine criterion:
Given $f\colon D\to\mathbb{R}, \lim\limits_{x\to a} f = A\Leftrightarrow \forall (x_n)\to a\colon x_n\neq a \Rightarrow f(x_n)\to A$
In words: A is a limit of f as x approaches a if and only if for every sequence $(x_n)$ converging to a such that $x_n\neq a$ their respective function f values converge to A.

EDIT: tiny error correction

 

[Stephen Tashi]

Is it theorem 5.15 in these notes ?: https://www.math.ucdavis.edu/~emsilvia/math127/chapter5.pdf

It's a familiar theorem in the study of metric spaces, but don't recall it being given a special name.

 

[nuuskur]

In our lectures it's specifically referred to as the Heine criterion, I just find it odd that googling this yields no result. A bit similar when I tried to look up a lemma given in our lectures as "Kuratowski-Zorn lemma", in most sources I just found it as "Zorn's lemma" which is curious.. 1st and 2nd world conflicts from the "good" ol' days?

Assuming you mean 5.1.15 in the notes, then yes, it does look the same. Thanks, I now know what to look for.

 

[homeomorphic]

Sequential criterion for limits.

 

[CellCoree]

The theorem you are referring to is known as the Heine criterion for the limit of a function. It states that the limit of a function exists at a point if and only if the function approaches the same value for every sequence approaching that point. This criterion is a useful tool in analyzing the behavior of functions and determining their limits. While it may be difficult to find information on this theorem in your native language, there is a wealth of resources available in English that can help you understand and utilize this important concept in mathematics.