Uniform continuous proof.

Question

If $f(x)$ define as: $f(x)=x\cos(\log(x))$, when $x>0$. $f(x)=0$, when $x=0$. we need prove that f is uniform continuous in $[0,+∞]$?

I already prove the following equation $|f(x)−f(y)|=|x−y||f′(c)| ≤2|x−y|$. I just feel confuse with the $δ=ϵ$ conclusion and how to write it.

Answer 1

You only need to to complete the last step:

For every $\epsilon>0$, we may set $\delta=\epsilon/2$. As we can see, if $|x-y|<\delta$, then $$|f(x)-f(y)|\leq 2|x-y|<2\delta=2\cdot\frac{\epsilon}{2}=\epsilon.$$