Sobel method on data points

Question

From what I've seen of the Sobel method, one takes an source image $A$, and applies the matrices $G_x = \begin{pmatrix} -1 && -2 && -1 \\ 0 && 0 && 0 \\ 1 && 2 && 1 \end{pmatrix} \star A$ and $G_x = \begin{pmatrix} -1 && 0 && 1 \\ -2 && 0 && 2 \\ -1 && 0 && 1 \end{pmatrix} \star A$, where $\star$ represents a convolution. My question is, what exactly is "$A$"? Is it represented by the coordinates of the a pixel? If so, given that one has an image in $2d$, then the two matrices above can't act on it; so that can't be right. In articles I've read, it often says that one acts on an "image function" $I(x,y)$. But what does this function look like; does it have a well-defined form?