I am studying the chapter on Degree theory and homotopy invariance theorem in the context of solving non-linear systems of equations from Iterative Solution of Nonlinear Equations in Several Variables, by Ortega and Rheinbolt.
My problem is the following:
Q1) Is degree theory and homotopy not applicable for real-valued maps ?
I ask because if I consider $f(x) = x+5, g(x) = x^2 + 5$ on $D = (-10, 10)$ then $\mathrm{deg}(f, D, 0) = 1$ but $\mathrm{deg}(g, D, 0) = 0$ since it has no root.
However, the function $H(x, t) = (1-t)*f(x) + t*g(x)$ satisfies the hypothesis of continuity, differentiablity, no solution on $\partial D$ etc, and application of homotopy invariance then gives $\mathrm{deg}(f, D, 0) = \mathrm{deg}(g, D, 0)$ which is clearly absurd.
Q2) What's wrong with my degree calculation or application of homotopy invariance ?