Monotony of Real Functions

Monotony

Definition: Monotony

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $S\subseteq \mathcal{D}$.

We say that $f$ is:

• increasing on $S$ if $x_1<x_2⟹f(x_1)\le f(x_2)$ for all $x_1,x_2\in S$;
• strictly increasing on $S$ if $x_1<x_2⟹f(x_1)<f(x_2)$ for all $x_1,x_2\in S$;
• decreasing on $S$ if $x_1<x_2⟹f(x_1)\ge f(x_2)$ for all $x_1,x_2\in S$;
• strictly decreasing on $S$ if $x_1<x_2⟹f(x_1)>f(x_2)$ for all $x_1,x_2\in S$.

If any of the above four cases, we also say that $f$ is monotone.

If $S=\mathcal{D}$, then we can omit the “on $S$” part.

Theorem: Monotony Criteria

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $[a;b]\subseteq \mathcal{D}$ be a closed interval.

If $f$ is differentiable on the open interval $(a;b)$, then:

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

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

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

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

PROOF

TODO

Theorem: Bijectivity of Real Monotonous Functions

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

If $f$ is continuous and strictly increasing or strictly decreasing, then $f$ is a bijective between $\mathcal{D}$ and its image $f(\mathcal{D})$.

PROOF

TODO

Theorem: Inverses of Strictly Monotonous Real Functions

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

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

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

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $[a;b]\subseteq \mathcal{D}$ be a closed interval.

If $f$ is monotone on $[a;b]$, then $f$ is also Riemann-integrable on $[a;b]$.

PROOF

TODO