Obliv

does this define a bijection from $\mathbb{R}$ to $(0,\infty)$? Let $f: \mathbb{R}\to (0,\infty)$ be defined by $$\begin{cases}
-\frac{1}{x} & x\leq-1 \\
0 & x=0 \\
2+x & -1<x \\
\end{cases}$$

In the context of this question: A vector field has the form $\vec{A} = -r\sin\theta\hat{i}+r\cos\theta\hat{j}$, where $r$ is the radius from $(0,0)$ and $\theta$ is the angle from $x$-axis, i.e. $r = \sqrt{x^2+y^2}$ and $\tan\theta = y/x$ and take on the usual polar coordinates.
a) What is the line integral of a $\vec{A}$ on closed path formed by a circle of radius $R$ moving counter-clockwise?