Understanding a limiting procedure
Say I have a function $f(x,y)$ that is defined for all $x \leq 0$ except at a single point $x=x_0 \neq 0.$ There exists one representation of the function in the domain $x < x_0$ and another representation of the function for $x_0< x \leq 0.$
Now consider another function $g(y)$ which I have and would like to see as the limit of $f(x,y)$ as $x \rightarrow 0.$
My question is:
If I take the limit as $x \rightarrow 0$ of the representation of the function $f(x,y)$ in the interval $x<x_0,$ would I expect to recover $g(y)$? My thought was initially no because the limit of $x \rightarrow 0$ is not in this representation’s domain of validity but I seem to obtain the correct result.
That is,
$$\lim_{x\rightarrow 0} \,\,f(x,y)|_{x<x_0} = g(y).$$
Am I being naive or is there a simple reason for that?
(This is something that came up in a physics project)
Thanks!
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