Is a domain of continuous function always an interval?

Question

MIT OCW states http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf page 3

"We say a function is continuous if its domain is an interval, and it is continuous at every point of that interval.

Example 5. The function $1/x$ is continuous on $(0, ∞)$ and on $(−∞, 0)$, i.e., for $x > 0$ and for $x < 0$, in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namely $x = 0$, and it has an infinite discontinuity there"

So it claims that $f: \mathbb{R} \backslash \{0\} \to \mathbb{R}$, $f(x)=1/x$ is not a continuous function because $\mathbb{R} \backslash \{0\}$ is not an "interval"

Then does it say there cannot exist any continuous function on a disconnected domain? Furthermore, I think, in MITOCW's view, any function cannot be continuous on domains containing isolated points

But I think that in the general topological space $X$ & $Y$, a $f:X \to Y$ is a continuous function iff for each open subset $V$ of $Y$, the set $f^{-1}(V)$ is an open subset of $X$. This condition doesn't require any formal restrictions on a domain.

So I think $f(x)=1/x$ and even sequences (assuming domain $N$ with subspace topology of usual topology) are continuous functions.

Also I found another question here

Uniformly continuous function on a disconnected domain

that makes mention of "uniformly continuous function on disconnected domain" which might assume existence of continuous function whose domain is not an interval.

Am I right? or wrong. If my conception of 'continuity' is wrong, please correct my arguments

Yes, the definition of continuity in the notes is nonstandard as, like you say, it disagrees with the topological definition of continuity of functions from subspaces of $\Bbb R$ to $\Bbb R$ (both endowed with the standard topology). — Travis Willse, Oct 25 '15 at 11:05
It's also not standard terminology to say that the function is discontinuous at $x=0$. See the discussion here, for example: http://math.stackexchange.com/questions/1087623/is-function-f-mathbb-c-0-rightarrow-mathbb-c-prescribed-by-z-rightarrow. — Hans Lundmark, Oct 25 '15 at 11:13
Does this answer your question? What is the formal definition of a continuous function? — FShrike, Sep 03 '22 at 07:28

IV_ · Answer 1 · 2022-11-14T20:32:20.450

Wikipedia - Continuous function:
"There are several different definitions of (global) continuity of a function, which depend on the nature of its domain.
A function is continuous on an open interval if the interval is contained in the domain of the function, and the function is continuous at every point of the interval. A function that is continuous on the interval ($-\infty,+\infty$) (the whole real line) is often called simply a continuous function; one says also that such a function is continuous everywhere.
...
Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples are the functions $x\mapsto\frac{1}{x}$ and $x\mapsto\tan ⁡ x$. When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere."

A function is called continuous if it is continuous at every point of its domain.

Continuity can not only be defined for intervals, but also, for example, for functions of several variables, for domains and for topological spaces.

A function between two topological spaces is continuous if and only if the pre-images of open sets are open sets.
A function between two topological spaces is continuous if and only if the pre-images of closed sets are closed sets.

Is a domain of continuous function always an interval?

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