 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 [...].