Can a sequence be uncountably infinite? The wikipedia article for a sequence defines it as such:

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. [...] The number of elements (possibly infinite) is called the length of the sequence. [...] Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. [...]

The three parts that I've marked give the impression that cardinality of a sequence as a collection of elements can be infinite but it can only be countably infinite because it is "enumerated" and also it is "defined as a function from natural numbers".
While this makes intuitive sense to me, it is in contrast with the accepted answer to this question where it states that any sequence in the form of $f:\mathbb{N}\rightarrow\mathbb{R}$, like $f(n)=n^2$, is "always continuous". This is odd to me because even though the codomain of $f$ is continuous, its domain and ultimately its cardinality is discrete. This can be demonstrated by writing down the elements of $f$: $1, 4, 9, 16, 25, ...$; one can see that while all element are members of $\mathbb{R}$, they arise from a calculation carried out on members of $\mathbb{N}$ and therefore end up constituting a discrete collection of numbers that is not continuous in any way.
The accepted answer to this question as well, states that sequences are continuous.
My question is, are sequences continuous or discrete? are they always countable or can they be uncountable?
P.S.: I would like to also ask a more general and seemingly unrelated question: is the continuousness of a function determined by that of its domain or codomain?
 A: Sequences in a set $X$ are functions $\Bbb N\to X$. The image therefore cannot have cardinality greater than that of $\Bbb N$, so sequences are always countable (or finite: the meaning of "countable" can vary). "Discrete cardinality" doesn't mean anything as far as I know.
As functions $\Bbb N\to\Bbb R$ with the usual topology on both sides (discrete, Euclidean) we get that all sequences are continuous functions because all functions out of a discrete space are continuous, in the sense of topology. However it's quite common to discuss whether or not a sequence has any accumulation or limit points, and I think you're confusing ideas around this with "continuity".
It does not mean anything to talk about a "continuous (co)domain". It does make sense to talk about a "discrete (co)domain", but the adjective 'continuous' must be used with great caution. It means one very precise thing that newcomers to topology/analysis often blur with more intuitive notions of 'smoothness'.
A: In analysis, the term "sequence" usually means a function from $\mathbb N$ to $\mathbb R$. Of course, people are free to use the term "sequence" in another way, e.g. to refer to a function from $\mathbb N$ to $\mathbb C$ (a "complex-valued sequence"), or a function from $\mathbb Z$ to $\mathbb R$ (a "bi-directional sequence"). It is normally clear from context what definition someone has in mind, or it could be that the precise definition of a sequence does not matter.
However, I suspect that your confusion does not stem from the definition of a sequence. In fact, the terms being misused here are "continuous" and "discrete".
If $f$ is a real-valued function (i.e. its domain and codomain are both subsets of $\mathbb R$), and $a$ is in the domain of $f$, then $f$ is said to be continuous at $a$ if for every $\varepsilon>0$ there is a $\delta>0$ such that, for all $x$ in the domain of $f$, if $|x-a|<\delta$ then $|f(x)-f(a)|<\varepsilon$. If $X$ is a subset of the domain of $f$, then $f$ is said to be continuous on $X$ if $f$ is continuous at every point of $X$. Simply saying that $f$ is continuous means that $f$ is continuous on its domain.
While it makes sense to speak of a function being continuous, there is no standard definition of a set being continuous. And, as Thomas Andrews mentions in the comments, if we were to define "continuity" for sets, this definition would bear no relation to the definition for functions. However, using the above definition, we can make sense of the term "continuous sequence": it is a function from $\mathbb N$ to $\mathbb R$ which is continuous.
A subset $S$ of $\mathbb R$ is discrete if for every $x\in S$, there is a $\delta>0$ such that $(x-\delta,x+\delta)\cap S=\{x\}$, i.e. the only point in $S$ which is less than $\delta$ away from $x$ is $x$ itself. There is no standard definition of the cardinality of a set being discrete: the cardinality of a set is a number, not a subset of $\mathbb R$.
Since $\mathbb N$ is discrete, and every real-valued function with a discrete domain is continuous, it follows that every sequence is continuous. Therefore, we don't usually speak of "continuous sequences", just as we don't usually speak of "triangles with three sides".
