Prove or disprove that: the function $ f : \mathbb R \to \mathbb R $ defined by $ f(x) := x^{1/3} \log (1+|x|)$ is uniform continuous!
I have tried all possible (I THINK) to prove it to be uniform continuous but failed. I think it would be NOT uniform continuous!Now, here I am still unable to find suitable sequences having their distance going to zero, BUT the distance between their functional images converging to some non-zero!
Hint: Prove that its derivative is bounded.