Connecting highschool continuity into undergraduate continuity

Question

In highschool, I learned continuity of a function $f: \mathbb{R} \to \mathbb{R}$ at a point $a$ to be defined as the following condition holding:

$$ \lim_{x \to a} f(x) = f(a) \tag{1}$$

However in undergraduate analysis book, the continuity is defined as so. A function $f: \mathbb{R} \to \mathbb{R}$ at a point is continous, if for every $\epsilon>0$, there exists a $\delta$ such that:

$$ |x-a | < \delta \implies |f(x) - f(a) | < \epsilon \tag{2}$$

In this video by Michael Penn , it's noted that somehow the undergraduate definition is more general although it is not explained precisely how. I think that they are actually the same. I think so clearly (1) implies (2) [take definition of limit], but I am having difficulty understanding why (2) doesn't imply (1). Could someone explain abstractly what the issue is? ( Examples are helpful but please don't make the entire explanation based on that, thanks)

Answer 1

Whether $(1)$ and $(2)$ are actually different depends on how you define $\lim_{x\to a}f(x)$.

There are two typical ways to define it, one is so called $\varepsilon-\delta$ definition and the other uses sequences.

Def. 1. Let $L$ such that $$(\forall\varepsilon>0)(\exists \delta > 0 )\quad 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon.$$ Then we write $L = \lim_{x\to a}f(x)$.

Def. 2. Let $L$ such that for every sequence $(x_n)$ ($x_n\neq a$) that converges to $a$ we have that sequence $(f(x_n))$ converges to $L$. Then we write $L = \lim_{x\to a}f(x)$.

When we write $\lim_{x\to a} f(x) = f(a)$, if we use def. 1., we say that $f$ is continuous at point $a$. If we use def. 2., we say that $f$ is sequentially continuous at point $a$. It is a well known theorem that for real functions of a real variable these two definitions coincide, so definitions of continuity $(1)$ and $(2)$ are actually equivalent in this case.

However, there are general types of spaces where limits and continuity make sense called topological spaces. In such spaces every continuous function is sequentially continuous, but sequentially continuous function doesn't need to be continuous. The reason for this is that topological spaces can be complicated and sequences alone might not be enough to describe behaviour around a single point. In that case, instead of sequences we need generalized sequences called nets or hypersequences. The main distinction is that sequences are indexed by a countable set, while nets can be indexed by uncountable sets.

Answer 2

If we're talking about a real-valued function defined on $\Bbb R$ the two are the same. Maybe point in that video is that (1) makes sense in the context of topological spaces, where there's no such thing as $|x-a|$.

In fact, in the context of topological spaces, (1) is equivalent to the standard definition of continuity. I've been asked why.

Say $X$ and $Y$ are topological spaces and $f:X\to Y$.

Def. Given $a\in X$ and $b\in Y$ we say that $\lim_{x\to a}f(x)=b$ if for every neighborhood $V$ of $b$ there exists a neighborhood $U$ of $a$ with $f(U)\subset V$.

Def. $f$ is continuous if the inverse image of any open set is open.

Prop. $f$ is continuous if and only if (1) holds for every $a\in X$.

Proof. Suppose first that $f$ is continuous and $a\in X$. Let $V$ be a nbd of $f(a)$. Now $U=f^{-1}(V)$ is a nbd of $a$ with $f(U)\subset V$.

Otoh suppose (1) holds for every $a\in X$ and let $V\subset Y$ be open. Let $U=f^{-1}(V)$; we need to show that $V$ is open. For each $a\in U$ there exists an open set $U_a$ with $a\in U_a$ and $f(U_a)\subset V$. Hence $U_a\subset U$. So $U=\bigcup_{a\in U}U_a$, hence $U$ is open.

Answer 3

I can't read the mind of Michael Penn, but here is one case where the undergraduate definition is needed: A domain with isolated points.

Say, for instance, a function defined on the integers. An expression like $\lim_{x\to a}f(x)$ (more commonly written as $\lim_{n\to a}f(n)$) makes no sense for finite $a$, so the highschool definition can't be used to even ask the question about continuity. But the undergraduate definition still lets you probe whether the function is continuous (and in this particular case you will find that all functions are continuous).

It is true that for functions with, say, some Euclidean space $\Bbb R^n$ as domain (and standard metric / topology), the two notions are equivalent. And even more generally, whenever both notions of continuity make sense, they will agree on the question of continuity. But for general metric spaces (with a trivial rewrite $|x-a|\mapsto d(x, a)$), the undergraduate definition is the only one of the two that makes sense everywhere.

(Do note that it goes the other way too: there are domains where limits make sense but distances don't, so there are times where the highschool definition works but the undergraduate definition doesn't. So I wouldn't say that either is more general than the other.)