Real Analysis: Proving f is a Constant Function

[mglaros]

Homework Statement

Let f be any function on the real line and suppose that: |f(x)-f(y)|<=|x-y|^2 for all x,y in R. Prove that f is a constant function. Note: "<=" reads "less than or equal to"

Homework Equations

The Attempt at a Solution

I have tried proof by contradiction, it seems to be the most obvious route in proving this statement. I started by assuming that there exists x,y in the domain of the function f(x) such that f(x) is not equal to f(y). I wasn't really able to proceed much further from there. Any help towards finishing this proof or perhaps a different approach would be greatly appreciated.

 

[lanedance]

can you use derivatves?

what is the magnitude of the derivative as x->y?

 

[Mark44]

can you use derivatves?
what is the magnitude of the derivative as x->y?

It's not given that the function is even continuous, let alone differentiable, so we can't assume that f' exists.

 

[Dick]

It's not given that the function is even continuous, let alone differentiable, so we can't assume that f' exists.

You could try and prove f' exists. Or you could try and split the interval between x and y into 2^n parts and see what the inequality tells you as n->infinity.

 

[lanedance]

fair bump, I thought uniform continuity & differentiablilty would follow reasonably easy from the definition, though i do like Dick's 2nd suggestion

 

[jgens]

Spliting the interval into n equal parts should suffice.

 

[jgens]

How's this?

Let $0<|x-a|<\mathrm{min}(1,\varepsilon)$. From this, we have that $0\leq|f(x)-f(a)|\leq|x-a|^2<|x-a|<\varepsilon$. This proves that $f$ is continuous.

From the defining property of $f$, we know that,

$$\frac{|f(x)-f(a)|}{|x-a|} \leq |x-a|$$

Since $f$ is continuous, evaluating the limit as $x \to a$, we find that $f'(a) = 0$ which proves that $f$ is a constant function.

I realize this is really rough, but could this approach be used to prove the initial problem? It's real late nowso I'm sure that it's riddled with errors.

 

[snipez90]

Well yes for this you can show by definition that |f'(a)| = 0 for every a due to mean value theorem. You can run into troubles if you tried applying this to say, finding isometries on R. Obviously you only need to show differentiability, but anytime you see |f(x)-f(y)| bounded by something involving |x-y|, it's going to be continuous.

 

[Quantumpencil]

I think your proof is fine. I remember doing this problem last year and I think that's basically the solution I used; there may have been some subtlety I overlooked, but I don' think so.