Why is the arctangent function restricted to the first and fourth quadrants?

Answer 1

A function cannot be one to many.

Suppose y = tan(x)

Now, since tan(x) = tan(x + pi)

then tan(x + pi) = y

But that means arctan(y) can be x or x+pi

In order to prevent that sort of indeterminacy, the arctan function must be restricted to an interval of width pi.

Any interval of that width would do and it could have been restricted to the first and second quadrants, or even from -pi/4 to 3*pi/4. The problem there is that in the middle of that interval the tan function becomes infinite which means that arctan would have a discontinuity in the middle of its domain. A better option, then, is to restrict it to the first and fourth quarters. Then the asymptotic values occur at the ends of the domain, which leaves the function continuous within the whole of the open interval.