1. ## monotonic functions

Let f be defined on an interval I and suppose that f is one to one on I.

(a) Give an example to show that f may not be monotone on I.
(b) Give an example to show that f may not be monotone on any subinterval of I.
(c) Suppose that f is continuous on I. Prove that f is monotone on I.
(d) Suppose that f has the intermediate value property on I. Prove that f is monotone on I.

2. Originally Posted by friday616
Let f be defined on an interval I and suppose that f is one to one on I.

(a) Give an example to show that f may not be monotone on I.
(b) Give an example to show that f may not be monotone on any subinterval of I.
(c) Suppose that f is continuous on I. Prove that f is monotone on I.
(d) Suppose that f has the intermediate value property on I. Prove that f is monotone on I.
For (a) and (b) let $f(x)= x$ if $x \in I \cap \mathbb{Q}$ and $f(x)=-x$ if $x \in I \cap (\mathbb{R} - \mathbb{Q})$. For (d) suppose $x and $f(x) (other possible cases are similar) and suppose there is a $z \in (x,y)$ such that $f(z) then by the intermediate value property there is a $c \in (z,y)$ such that $f(c)=f(x)$ so $f$ is not one-one. Similarly there is no $z \in (x,y)$ such that $f(z)>f(y)$ and so for all $z \in (x,y)$ $f(x). Now pick any $a,b \in I$ $a and show that $f(a) (Consider the cases where $(a,b)$ and $(x,y)$ are disjoint and when they're not separately). Since a continous function satisfies the ivp, you're done.