Consider the following conditions on a function $f$ whose domain is the closed interval $[0,1]$. (For any condition involving a limit, at the endpoints, use the relevant one-sided limit.) I. $f$ is differentiable at each $x \in [0,1]$. II. $f$ is continuous at each $x \in [0,1]$. III. The set $\{f(x) \mid x \in [0,1]\}$ has a maximum element and a minimum element. Statements (13) If I is true, then II is true. (14) If II is true, then III is true. (15) If III is false, then I is false. (16) No two of the three given conditions are equivalent to each other. (Two statements being equivalent means each implies the other.)
Consider the following conditions on a function $f$ whose domain is the closed interval $[0,1]$. (For any condition involving a limit, at the endpoints, use the relevant one-sided limit.)\\
I. $f$ is differentiable at each $x \in [0,1]$.\\
II. $f$ is continuous at each $x \in [0,1]$.\\
III. The set $\{f(x) \mid x \in [0,1]\}$ has a maximum element and a minimum element.
Statements
(13) If I is true, then II is true.\\
(14) If II is true, then III is true.\\
(15) If III is false, then I is false.\\
(16) No two of the three given conditions are equivalent to each other. (Two statements being equivalent means each implies the other.)