Does $\delta$ have to be less then $\epsilon$ for uniform continuity?

Question

Does $\delta$ have to be less than $\epsilon$ for uniform continuity? If not, what is the implication of $\delta < \epsilon$ in a uniform continuity proof?

Answer 1

You could add this assumption since if $\delta_1$ is good for uniform continuity then so is every $0<\delta_2<\delta_1$. However "the best you can do" mean finding the biggest values of $\delta$ for which the condition of uniform continuity is satisfied and this $\delta$ may be and may be not smaller than $\epsilon$: just think of functions $f(x)=ax$ for $a<1$ and $a>1$ (say, positive).