# Definition: Monotonic Functions

Based on the order relation for real numbers, real functions $$f:\mathbb R\mapsto\mathbb R$$ can be classified by the way the values of $$f(x)$$ increase (decrease), depending on increasing $$x$$. The function $$f$$ is called:

monotonically increasing, if for all $$x$$ and $$y$$ such that $$x \le y$$ one has $$f(x) \le f(y)$$,

strictly monotonically increasing, if for all $$x$$ and $$y$$ such that $$x < y$$ one has $$f(x) < f(y)$$,

monotonically decreasing, if for all $$x$$ and $$y$$ such that $$x \le y$$ one has $$f(x) \ge f(y)$$,

strictly monotonically decreasing, if for all $$x$$ and $$y$$ such that $$x y$$ one has $$f(x) > f(y)$$.

If $$f$$ is either (strictly) monotonically increasing or decreasing, it is called monotonic.

Definitions: 1
Lemmas: 2
Proofs: 3 4 5 6 7 8 9
Propositions: 10 11 12 13 14 15 16 17 18 19 20
Sections: 21

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983