Let $f: U \to \mathbb{R}^{n}$ $C^{1}$ injective where $U$ is a open in $\mathbb{R}^{n}$ (so $f$ is open by invariance domain theorem).
a) Is there exist $f$ such that dim $ker(df_{x}) >$ dim $im(df_{x})$ for all $x \in U$ ?
b) Is there exist $f$ such that dim $ker(df_{x}) <$ dim $im(df_{x})$ for all $x \in U$ ?
Thank you