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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'.

TODO

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