- Thread starter
- #1
- Apr 13, 2013
- 3,844
Hey!!! 
I have a question..I am given the following exercise:
Let $f:[0,+\infty) \to \mathbb{R}$ uniformly continuous at $[0,+\infty)$.Prove that there are $a,b \geq 0$ such that $|f(x)| \leq ax+b , \forall x \geq 0$.
That's what I did so far:
$f:[0,+\infty) \to \mathbb{R}$ is uniformly continuous at $[0,+\infty)$.
So: $\forall \epsilon>0 \exists \delta>0 $ such that $\forall x,y \in [0,+\infty)$ with $|x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon$..
But,how can I continue to show that there are $a,b \geq 0$ such that $|f(x)| \leq ax+b , \forall x \geq 0$??

I have a question..I am given the following exercise:
Let $f:[0,+\infty) \to \mathbb{R}$ uniformly continuous at $[0,+\infty)$.Prove that there are $a,b \geq 0$ such that $|f(x)| \leq ax+b , \forall x \geq 0$.
That's what I did so far:
$f:[0,+\infty) \to \mathbb{R}$ is uniformly continuous at $[0,+\infty)$.
So: $\forall \epsilon>0 \exists \delta>0 $ such that $\forall x,y \in [0,+\infty)$ with $|x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon$..
But,how can I continue to show that there are $a,b \geq 0$ such that $|f(x)| \leq ax+b , \forall x \geq 0$??
