# Homework Help: Calculus Proof

1. Oct 19, 2009

### ~Sam~

1. The problem statement, all variables and given/known data

Suppose that |f (x) - f (y)|  (x-y)^2 for all real numbers x and y: Prove
that f is a constant function.

2. Relevant equations

No relevant equations..

3. The attempt at a solution

I'm really stuck..i'm thinking you're suppose to use mathematical induction.?

2. Oct 19, 2009

### Dick

Can you show that |f(x)-f(y)|<=(x-y)^2 for all x and y implies that f is differentiable, and that it's derivative equals zero everywhere?

3. Oct 19, 2009

### lurflurf

Dick's approach is popular, but I never liked it. I do not think this problem merits the use of the derivative. I like to show inductively that
|f(x)-f(y)|<=(x-y)^2 implies
|f(x)-f(y)|<=(x-y)^2/2^n for any natural number n from which the result is obvious.
hint
|f(x)-f(y)|=|[f(x)-f((x+y)/2]+[f((x+y)/2)-f(y)]|<=|f(x)-f((x+y)/2|+|f((x+y)/2)-f(y)|

4. Oct 19, 2009

### Dick

That is a nice alternative approach. There's more than one way to skin a cat.