Which of the following statements are true?

I. Every strictly increasing function on $I$ is increasing on $I$.

II. Every increasing function on $I$ is strictly increasing on $I$.

III. If $f$ is strictly increasing on $I$, then $-f$ is strictly decreasing on $I$.

IV. If $f$ is strictly increasing on $I$, then $f(x) > 0$ on $I$.

Explanation

1. 01

   $f(x_1) < f(x_2)$ implies $f(x_1) \le f(x_2)$, so I is true; and multiplying by $-1$ reverses the inequality, so III is true.

2. 02

   A constant function is increasing but not strictly, so II is false; $f(x) = x$ is strictly increasing on $\mathbb{R}$ yet negative for $x < 0$, so IV is false.