Q:

What does it mean by "function defined near the origin"?

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.

A:

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.

