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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 monotone increasing iff \( f(y) \ge f(x) \) whenever \( x,y \in X \) and \( y > x \).

We say that \( f \) is strictly monotone increasing iff \( f(y) > f(x) \) whenever \( x,y \in X \) and \( y > x \).

We say that \( f \) is monotone decreasing iff \( f(y) \le f(x) \) whenever \( x,y \in X \) and \( y > x \).

We say that \( f \) is strictly monotone decreasing iff \( f(y) < f(x) \) whenever \( x,y \in X \) and \( y > x \).

We say that \( f \) is monotone iff \( f \) is monotone increasing or monotone decreasing.

We say that \( f \) is strictly monotone iff \( f \) is strictly monotone increasing or strictly monotone decreasing.

Some properties of monotic functions

  • Function continuity implies monotonicity? No
  • Function monotinicity implies continuity? No
  • Monotone functions on a closed interval obey the maximum principle (with continuity requirement ignored)? Yes
  • Monotone functions on a closed interval obey the intermediate value principle (with continuity requirement ignored)? No
  • If a function is strictly monotone and continuous, then one very nice property is that it is invertible.

Strictly monotone and continuous implies invertable

Let \( a < b \) be real numbers, and let \( f: [a, b] \to \mathbb{R} \) be a function which is both continuous and strictly monotone increasing.

Then \( f \) is a bijection from \( [a, b] \) to \( [f(a), f(b)] \), and the inverse \( f^{-1}: [f(a), f(b)] \to [a, b] \) is also continuous and strictly monotone increasing.

The same can be said replacing 'increasing' with 'decreasing'.

Example

TODO


Source

p242