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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 [...].