Date modified: Tue 2023-06-20 . 01:12 PM
Date created: Sat 2023-10-07 . 09:02 PM
# Some analysis problems. ![[---images/---assets/---icons/question-icon.svg]] Show that a real-valued continuous function $f$ on the real line is injective if and only if it is monotonic. ![[---images/---assets/---icons/question-icon.svg]] Show that a real-valued differentiable function $f$ on the real line is injective if and only if $f'(x) \le 0$ for all $x$ or $f'(x)\ge 0$ for all $x$, and that the set of vanishing derivatives $Z=\{x:f'(x)=0\}$ contain no intervals. ![[---images/---assets/---icons/question-icon.svg]] Show that the set of continuous functions $f:\mathbb{R}\to \mathbb{R}$ is equinumerous as $\mathbb{R}$. That is, the cardinality of the class $\mathscr C^0(\mathbb{R})$ is the continuum.