In R2\(axis X)the two nullhomotopic functions are not homotopic

Question

I want to prove that in space R2\X the two functions f&g are not homotopic if we define g(x)=c1& f(x)=c2 c1 and c2 are two points one of the above axis X and the othe one is under it

Answer 1

Fix a point $p$ in the domain of $f$ and $g$ and suppose there is a homotopy $H \colon D \times I \to \mathbb{R}^2 - X $ between $f$ and $g$. Then $H$ restricted to $\{p\} \times I $ is a path between $c_1$ and $c_2$, which is impossible since they lie in different path components of $\mathbb{R}^2 - X$.