When a line is not in the domain of a function. Let us suppose that $f(x,y)$ is a function and a line such as $x=a$ (where $a$ is any real number) is not in the domain of that function. What would that imply about the graph around a point such as $(a,b),$ where $b$ is also any real number?

It implies nothing; meaning, you cannot derive any useful information about what the function is doing "near" a point $(a,b)$ just from knowing that the function is undefined on the line $x=a$.
This is just like in $1$-variable: if I tell you that I have a function $y=f(x)$, and that it is undefined at $x=a$, what does that tell us about the behavior of the function near $x=a$ (that is, what does that tell us about $\lim\limits_{x\to a}f(x)$)? Answer: The information provided is insufficient to answer the question meaningfully; the fact that the function is undefined at $x=a$ does not, in and of itself, tell us anything about what the function does near $a$.
And the same is true for a function of two variables that is undefined on the line $x=a$. The only thing that it tells us about the behavior near $(a,b)$ is that it is undefined at the points $(a,b')$ for all $b'$ near $b$; in and of itself, however, it does not tell us anything about the behavior or values of the function at other points near $(a,b)$ (or far form $(a,b)$, for that matter).
