Definition Let $A$ be a subset of $\mathbb{R}$, and let $f$ be a function from $f:A\to\mathbb{R}$. Then

1. 1.

   $f$ is increasing or weakly increasing, if $x\leq y$ implies that $f(x)\leq f(y)$ (for all $x$ and $y$ in $A$).

2. 2.

   $f$ is strictly increasing or strongly increasing, if $x<y$ implies that $f(x)<f(y)$.

3. 3.

   $f$ is decreasing or weakly decreasing, if $x\leq y$ implies that $f(x)\geq f(y)$.

4. 4.

   $f$ is strictly decreasing or strongly decreasing if $x<y$ implies that $f(x)>f(y)$.

5. 5.

   $f$ is monotone, if $f$ is either increasing or decreasing.

6. 6.

   $f$ is strictly monotone or strongly monotone, if $f$ is either strictly increasing or strictly decreasing.

Theorem Let $X$ be a bounded or unbounded open interval of $\mathbb{R}$. In other words, let $X$ be an interval of the form $X=(a,b)$, where $a,b\in\mathbb{R}\cup\{-\infty,\infty\}$. Futher, let $f:X\to\mathbb{R}$ be a monotone function.

1. 1.

   The set of points where $f$ is discontinuous is at most countable [1, 2].

2. Lebesgue

   $f$ is differentiable almost everywhere ([3], pp. 514).