Are the following true? If so, how does one prove them?
Suppose $f:\Bbb R\to\Bbb R$ is continuous.
(i) $\partial \{f>t\} \subset \{f=t\}$, where $\{f>t\}:=\{x \in\Bbb R:f(x)>t\}$ and $\partial$ denotes the boundary of the set.
(ii) The set $\{f=t\}$ has Lebesgue measure $0$ except for at most countably many $t$.