Does a vector have to be continuous to fall within a set? The question asks: explain why $\ f(x) = $ $\ x \over \ x^2 + 4x + 3$ is a vector in $C[0, 3]$ but not a vector in $C[-3, 0]$.
I know that $f$ is not continuous on $C[-3, 0]$ at $x = -1$ and $x = 3$. Does that mean for a (function) vector to fall in a certain set, it has to be continuous? I can't find a concrete definition or rule for this. Why does the function have to be defined for all values in the set?    

$C(X)$ is defined to be the set of continuous functions $X\rightarrow \mathbb{R}$.  
To say that something is a function from $X$ to $Y$ requires that it is defined at every element of $X$.
So saying that $f$ is a vector in $C[0,3]$ or $C[-3,0]$ is saying that it is a continuous map defined on $[0,3]$ and $[-3,0]$, respectively.
However, given a vector space $V$ whose elements are functions, unless specified otherwise (as is the case for $C(X)$) it is in general not necessary for those functions to be continuous.  For example, the vector space of all functions from a set $X$ to $\mathbb{R}$ need not even have `continuity' be a meaningful notion unless $X$ has been given a topology.
