Degree of a constant map $f : S^2 \to S^2$ Suppose we have a map $f : S^2 \to S^2 : x \mapsto a$ for some constant $a$. Then $f$ is not surjective, so $\deg f = 0$. 
But it is possible to define a homotopy $F$ between $f$ and the identity map: for each point $x$, define $F_(x,t)$ to travel along the great circle between $x$ and $a$. Then $\deg f = \deg \text{id} = 1$. This is a contradiction.
Where have I gone wrong? I assume $F$ is not really a homotopy, if so, can someone explain why?
(For context, I am working through Hatcher chapter 2.2.)

Your proposed homotopy $F(x, t)$ is not well-defined for $x = - a$, because there are many different great circles containing $a$ and its antipodal point $-a$. There is no way to choose one of these great circles that will make $F(x, t)$ continuous.
