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