MIT OCW states
http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf page3
"
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 $f:R/{0}->R$, $f=(1/x)$ is not a continuous function because $R/{0}$ 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 $f:X->Y$ 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 $f=1/x$ 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
Best Answer
Your citation "We say a function is continuous if its domain is an interval, and it is continuous at every point of that interval." is only a partial aspect of continuity. This sentence is not: "We say a function is continuous iff its domain is an interval, and it is continuous at every point of that interval."