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?
When a line is not in the domain of a function.
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calculus
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1F(y), or F(x)? Generally, we say $x=5$ is not in the domain of $y=f(x)$ to mean that the *point* $x=5$ is not a value that can be "plugged into" the function. – 2012-03-27
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0I apologize, because I have not asked what I wanted to ask nor have I made myself clear. 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)? – 2012-03-27
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0If you have a function of two variables with *real* values, then the "graph" is like a topography map in 3-dimensions (see, for instance, this [java grapher](http://www.calculator-grapher.com/graphers/function-grapher-2-var.html). Saying that $x=a$ is not in the domain means that there is no graph above that line (just like, when you have a function of one variable, if $x=k$ is not in the domain, then the graph has no point "above" the point $x=k$). – 2012-03-27
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0Please consider editing the body of the question and making yourself clear in it, instead of in the comments only. – 2012-03-27
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0If the line $x=a$ is not in the domain, then you cannot graph the function at $(a,b)$, just like you cannot graph $y=1/x$ at $0$. "near it", what it does depends on the function; it need not imply anything. The function $f(x,y) = xy/x$ is not defined on the line $x=0$, but the graph "near" points of the form $(0,b)$ is not in any way "weird". On the other hand, the graph of $f(x,y) = y/x$ is undefined on the line $x=0$, and it can "blow up" near points $(0,b)$ (to $\infty$ on one side and to $-\infty$ on the other, and do really weird things near $(0,0)$) – 2012-03-27
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0Thank you very much, but to probe the question a little deeper, what would happen to the limit around the point (a,b)? For example, suppose g(b)=0=f(a). What would happen to the limit as it proceeds to the point (a,b) on the graph f(x,y)=g(y)/f(x) and x=a is not in the domain of f(x,y)? – 2012-03-27
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0What happens to a one variable function near a point where it is undefined? Answer: It depends on the function. Exactly the same happens for two-variable functions. What happens near a point where it is undefined depends on the function; there are no general things to be said, deep or shallow. – 2012-03-27
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0See edit above. – 2012-03-27
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0See answer below: in and of itself, it tells us absolutely nothing other than what we already knew: it is undefined at points $(a,b')$ for any $b'$. – 2012-03-27