A specific question related to the Brouwer degree

Question

Let $\Omega \subset \mathbb{R}^n$ be open and bounded, plus the relative open $\Omega_1$ and $\Omega_2$ of $\Omega$. Taking functions $f \in C^1(\Omega) \cap C(\overline{\Omega})$ and $g \in C^2(\Omega) \cap C(\overline{\Omega})$ such what $$|f - g|_{\max} = \max_{x \in \overline{\Omega}} |f(x) - g(x)| < \mbox{dist}(y,f(\partial\Omega)),$$ where $f^{-1}(y) \neq \emptyset$ and $y \not\in f(\overline{\Omega} \setminus (\Omega_1 \cup \Omega_2))$. How can you conclude that $y \not\in g(\overline{\Omega} \setminus (\Omega_1 \cup \Omega_2))$ also occurs?