I need to prove that the following function in continuous.
Let C a circle with radius r. For each $y \in [-r,r]$, we associate the point $x \in C$ obtained by the intersection of C and the vertical line through y, in the superior plane. After getting x, we draw the radius containing this point and determine the central angle $\alpha$.
This function is clearly a bijection.
How to proove that this function is continuous? I have problems when using the definition of continuity because I can't conect linear measures in $\mathbb{R}$ and arc measures on the circle C.
Thanks a lot!