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My textbook says "for a function f(x) defined near the origin, a linear function p(x) = a + bx is called a linear approximation of f(x) near the origin".

By "a function f(x) defined near the origin", does it mean that the function is mapped only with values of x near the origin? Or does it mean that the author will only care about the ranges near the origin when talking about the function's linear approximation?

This may not be too important for my understanding the important concepts of linear approximations, but it has been bugging me the last few days so I had to ask.

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  • $\begingroup$ There is a conception "LIMITATION" in calculus. So, the near the origin means a neiborhood of orgin in the real line. $\endgroup$ – Brooks Jun 23 at 15:42
  • $\begingroup$ It simply means the function is defined in some neighbourhood of the origin (and possibly in a larger set). $\endgroup$ – Bernard Jun 23 at 15:42
  • $\begingroup$ Or more simply, the origin is an inner point of the domain of the function. $\endgroup$ – LutzL Jun 23 at 15:44
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It most often means that there is some $\epsilon >0$ such that function is defined in the interval $(-\epsilon,\epsilon)$ around the origin. Sometimes the interval can be closed interval $[a,0]$ or $[0,b]$ that has zero as endpoint.

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I would guess he means that the function is defined on an open neighborhood of the origin. What is meant by defined depends on context. It might mean that there are no singularities.

E.g. the function $f(x) = \frac{1}{x}$ is not defined near the origin of the real line, but the function $g(x) = \frac{1}{x + 1}$ is.

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Most often that means that the function is defined on a small intervall (often even chosen symmetric) around the origin.

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