A simple example of homotopy

Yes. Here, the most natural homotopy between $f$ and $g$ is simply the map $H(t,x)=(x+t)^{2}$.
It is continuous, and it verifies $H(0,x)=x^{2}=f(x)$, while $H(1,x)=(x+1)^{2}=g(x)$.