How can we generalize the linear equation $Ax=b$ from finite to uncountable dimensions? Starting with the equation $Af=g$ where $f,g \in$ $\mathbb{R}^2$ and $A$ is a $2$x$2$ matrix, suppose that we generalize the two indexes of $f=$ ($f$$1$, $f$$2$) to a continuum, e.g all real numbers, so that $f$ becomes a function $f:$ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$. And the same is done for $g$. So $f$ and $g$ are indexed by their uncountable domains of definition. My question is:
Is there a way to accordingly generalize the matrix $A$ so that the equation $Af=g$ remains meaningful?  
Many possibilities seem to arise, e.g. if $f$ is differentiable and g is the derivative of $f$, then it seems that A should perform differentiation. 
If A is somehow made to contain the differential $dx$, then integration seems to lurk behind. 
I know that functional analysis studies these ideas, but the books I know of start directly with abstract settings, so I wonder whether one organize this index-based viewpoint and build a natural bridge from the discrete to the continuous and from finite to infinite and uncountable dimensions. 

You would need a basis. And it's really a matter of convention whether one exists in general (you need the Axiom of Choice, in the form of Zorn's lemma). At any rate, it's impossible to actually construct / write down a concrete basis for spaces like the functions $\Bbb R\to\Bbb R$. And since you can't write down a basis, you can't write down a matrix representation of any linear transformations.
If you, for instance, limit yourself to just polynomials, then you have bases. The standard basis is $1,x,x^2,x^3,\ldots$, and under that basis, the linear transformation we call "differentiation" is given by the "matrix"
$$
\begin{bmatrix}0&1&0&0&\cdots\\
0&0&2&0&\cdots\\0&0&0&3&\cdots\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}
$$
