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