# Thread: Uniformly Continuous Problem

1. ## Uniformly Continuous Problem

Q: Let f:[0,infinity) be defined by f(x) = x^2/(x+1). Prove if it is uniformly continuous.

My solution: I let $\epsilon > 0, \delta > 0, x,y\epsilon[0,infinity)$ with $|x-y|<\delta$

then I have |f(x)-f(y)| = $|\frac{x^2}{x+1}-\frac{y^2}{y+1}|$ = $|\frac{x^2+x^2y-y^2x-y^2}{(x+1)(y+1)}|$

But I can't simpify it anymore to which would make the proof works...

Thank you.

2. Have you noticed that the derivative is bounded?
$f'(x) = \frac{{x^2 + 2x}}{{x^2 + 2x + 1}} \le 1\quad x \in [0,\infty ).$
Then we can get $\left| {f(x) - f(y)} \right| \le \left| {x - y} \right|$

3. I understand that the derivative is bounded, but how does that imply |f(x)-F(y)| < |x-y| ? I have not learn to do anything with derivative in my class yet.

4. Originally Posted by tttcomrader
I understand that the derivative is bounded, but how does that imply |f(x)-F(y)| < |x-y| ? I have not learn to do anything with derivative in my class yet.
Sorry to say, someone has gotten the cart before the horse!
This is a classical problem.
It is a Lipschitz Condition. The derivative is bounded.
That implies uniform continuity.
You need to confront your lecturer.