The definition of uniform continuity states that a function $f$ defined over an interval $I$ is uniform continuous if $\forall \epsilon > 0, \quad\exists \delta> 0$ such that $ \forall x' , x'' \in I : 0 < |x'-x''| < \delta \implies |f(x')-f(x'')|<\epsilon$ Is it possible to find multiple solutions for $\delta$ while using different approaches?
Thanks for the help!
Since we're after a $\delta$ such that$$|x-y|<\delta\implies\bigl|f(x)-f(y)\bigr|<\varepsilon,$$if you take $\delta'$ such that $0<\delta'\leqslant\delta$, then$$|x-y|<\delta'\implies|x-y|<\delta\implies\bigl|f(x)-f(y)\bigr|<\varepsilon.$$So, yes, of course that it is possible to have more than one $\delta$. In fact, there are always infinitely many $\delta$'s that will do.