Math and science::Analysis::Tao::09. Continuous functions on R
Monotonic functions
Let \( X \subseteq \mathbb{R} \) and \( f : X \to \mathbb{R} \) be a function.
We say that \( f \) is [...] iff \( f(y) \ge f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is [...] iff \( f(y) > f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is [...] iff \( f(y) \le f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is [...] iff \( f(y) < f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is [...] iff \( f \) is [...] increasing or [...] decreasing.
We say that \( f \) is [...] iff \( f \) is [...] monotone [...] or [...] monotone [...].
Some properties of monotic functions
- Function continuity implies monotonicity? [...]
- Function monotinicity implies continuity? [...]
- Monotone functions on a closed interval obey the maximum principle (with continuity requirement ignored)? [...]
- Monotone functions on a closed interval obey the intermediate value principle (with continuity requirement ignored)? [...]
- If a function is strictly monotone and continuous, then one very nice property is that [...].