Monotony of Real Functions

Monotony

Definition: Increasing Function

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is increasing if, for all $x_1,x_2\in D$,

$$x_1<x_2⟹f(x_1)\le f(x_2).$$

Definition: Strictly Increasing Function

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is strictly increasing if, for all $x_1,x_2\in D$,

$$x_1<x_2⟹f(x_1)<f(x_2).$$

Definition: Decreasing Function

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is decreasing if, for all $x_1,x_2\in D$,

$$x_1<x_2⟹f(x_1)\ge f(x_2).$$

Definition: Strictly Decreasing Function

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is strictly decreasing if, for all $x_1,x_2\in D$,

$$x_1<x_2⟹f(x_1)>f(x_2).$$

Definition: Monotonic Function

A real function is monotonic if it is (strictly) increasing or (strictly) decreasing.

Criteria for Monotony

Theorem: Monotony of Real Functions

Let $f:D\to \mathbb{R}$ be a real function which is continuous on a closed interval $D=[a;b]\subset \mathbb{R}$ differentiable on the open interval $(a;b)$:

• $f$ is increasing if and only if $f^{\prime }(x)\ge 0$ for all $x\in (a;b)$;

• $f$ is strictly increasing if and only if $f^{\prime }(x)>0$ for all $x\in (a;b)$;

• $f$ is decreasing if and only if $f^{\prime }(x)\le 0$ for all $x\in (a;b)$;

• $f$ is strictly decreasing if and only if $f^{\prime }(x)<0$ for all $x\in (a;b)$.

PROOF

TODO

Properties

Theorem: Bijectivity of Real Monotonous Functions

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

If $f$ is continuous and strictly monotonous, then $f$ is a bijective between $I$ and its image $f(I)$.

PROOF

TODO

Theorem: Inverses of Strictly Monotonous Real Functions

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

If $f$ is continuous and strictly increasing (strictly decreasing), then its inverse $f^{-1}:f(I)\to I$ is continuous and strictly increasing (strictly decreasing).

PROOF

Suppose that $f$ is strictly increasing. To prove that $f^{-1}$ is strictly decreasing, observe two $x_1,x_2\in I$ with $x_1<x_2$. Since $f$ is strictly increasing, we have $f(x_1)<f(x_2)$. Let $y_1=f(x_1)$ and $y_2=f(x_2)$, i.e. $y_1<y_2$. Therefore, $f^{-1}(y_1)=x_1$ and $f^{-1}(y_2)=x_2$ and so $f^{-1}(y_1)<f^{-1}(y_2)$ for $y_1<y_2$ which means that $f^{-1}$ is strictly increasing. The proof is analogous for when $f$ is strictly decreasing.

Theorem: Integrability of Monotone Functions

Every monotone real function on a closed interval is also Riemann-integrable on it.

PROOF

TODO