Topological definition of continuity and its application to epsilon-delta definition?

Question

So I am beginning Munkres' textbook on topology.

The topological definition of continuity reads:

$f:X\rightarrow Y$ is continuous if for each open subset $V\subset Y$, $f^{-1}(V)$ is an open subset of $X$.

Of course, it does fit the epsilon-delta definition of continuity since in

$\forall\epsilon\exists\delta,|x-x_0|<\delta\Rightarrow|f(x)-f(x_0)|<\epsilon$

$|x-x_0|<\delta$ and $|f(x)-f(x_0)|<\epsilon$ are both open.

Also, since openess of a set means closeness of that set's complement, the following epsilon-delta definition (which is valid) also agree with the topological definition:

$\forall\epsilon\exists\delta,|x-x_0|\le\delta\Rightarrow|f(x)-f(x_0)|\le\epsilon$

Yet my question is, how about the following epsilon-delta definition?

$\forall\epsilon\exists\delta,|x-x_0|\le\delta\Rightarrow|f(x)-f(x_0)|<\epsilon$

1.) Is it a valid definition of continuity of $\mathbb{R}^n$ regardless of the topological definition?

2.) If yes, does it agree with the topological definition of continuity?

Answer 1

You can consider 4 variations of the $\epsilon$-$\delta$ definition for continuity at a point $x_0$ in $\mathbb{R}^n$.

1. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon)$.
2. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | \le \delta \Rightarrow |f(x) - f(x_0)| < \epsilon)$.
3. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | < \delta \Rightarrow |f(x) - f(x_0)| \le \epsilon)$.
4. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | \le \delta \Rightarrow |f(x) - f(x_0)| \le \epsilon)$.

One can easily check that all of these are mutually equivalent, and we can even do this in any pair of metric spaces instead of $\mathbb{R}^n$. But this is a simple consequence of properties of the order in $\mathbb{R}$ and properties of $\mathbb{R}$ itself.

E.g. to see that 1 implies 2: assume 1 holds, and pick $\epsilon>0$. Then 1 guarantees us a $\delta>0$ with the properties of 1. and then for 2. we just pick $\delta' = \frac{\delta}{2} > 0$. Then if $|x - x_0| \le \delta' = \frac{\delta}{2} < \delta$, so 1 guarantees that indeed $|f(x) - f(x_0)| < \epsilon$.

Other implications go similarly.

And all of these are equivalent to the topological definition. The proof of which uses the first, the most standard, variant as this corresponds best to interior points / open balls.

(Proof in one direction: suppose $O$ is open in the image, and let $x_0$ be in $f^{-1}[O]$. Then $f(x_0) \in O$, so there is some open ball with radius $\epsilon>0$ such that $B(f(x_0), \epsilon) \subset O$. Apply definition 1 to this $\epsilon$ to get $\delta>0$, and then one checks that $B(x_0, \delta) \subset f^{-1}[O]$. The $<$ signs give us nice open balls, so the proof is "smooth")

But the $\le$ instead of $<$ does not have anything to do with inverse images of closed sets being closed etc. The 4 variants are mutually equivalent, and the first one is the more natural to prove the equivalence to the topological definition.