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Monotonicity of Real Functions#

Definition: Monotonicity

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 \(x_1 \lt x_2 \implies f(x_1) \le f(x_2)\) for all \(x_1, x_2 \in S\);
  • strictly increasing on \(S\) if \(x_1 \lt x_2 \implies f(x_1) \lt f(x_2)\) for all \(x_1, x_2 \in S\);
  • decreasing on \(S\) if \(x_1 \lt x_2 \implies f(x_1) \ge f(x_2)\) for all \(x_1, x_2 \in S\);
  • strictly decreasing on \(S\) if \(x_1 \lt x_2 \implies f(x_1) \gt f(x_2)\) for all \(x_1, x_2 \in S\).

In 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: Monotonicity 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 continuous on \([a;b]\) and differentiable on the open interval \((a;b)\), then:

  • \(f\) is increasing on \([a;b]\) if and only if \(f'(x) \ge 0\) for all \(x \in (a;b)\);
  • \(f\) is decreasing on \([a;b]\) if and only if \(f'(x) \le 0\) for all \(x \in (a;b)\);
  • \(f\) is strictly increasing on \([a;b]\) if \(f'(x) \gt 0\) for all \(x \in (a;b)\);
  • \(f\) is strictly decreasing on \([a;b]\) if \(f'(x) \lt 0\) for all \(x \in (a;b)\).
Proof

TODO

Theorem: Invertibility of Strictly Monotone Functions

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

If \(f\) is strictly monotone, then \(f\) is injective. Moreover:

Proof

TODO

Theorem: Riemann-Integrability of Monotone Functions

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

If \(f\) is monotone on a closed interval \([a,b] \subseteq \mathcal{D}\), then \(f\) is also Riemann-integrable on \([a, b]\).

Proof

TODO