TITLE: Intuitive and convincing argument that functions are vectors
QUESTION [9 upvotes]: Back to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge the gap between vectors and functions.
When I ask them why, the answer I most often get are as follows:

Vectors are something learned in linear algebra, and they are
basically pointy arrows in $R^2$ and written with brackets $\{\}$ or sometimes with
arrow on the top
Functions are completely different, they are the object under study in calculus, never drawn as an arrow, doesn't
satisfy some axioms of vector space, can be drawn in a lot of ways
(piecewise or continuous, with crazy oscillations). 
You can do a lot with functions, such as taking the derivative of it,
or the integral of it. You can find the inverse of a function.  
No authority has ever said functions are vectors, if they did I would
believe them
Yes, functions are sometimes contained in brackets, but that is just a
vector of functions, not a function

Can someone please provide a good example may bridge this gap? Also, is there any books that explicitly bridge the two concepts?
Thank you for your inputs!

REPLY [3 votes]: There is another way function could be viewed as a vector. For practical calculations, plotting, etc. functions are sometimes discretized, sufficiently finely. For example, $x\mapsto x^2$, $x\in[0,3]$ can be represented as its values on the nodes 0,1,2,3 as a 4-dimensional vector $(0,1,4,9)^T$.