/sci/ - Science & Math

JUST FINISHED TODAY

Stuck with a detail of the proof of [math]f:\mathbb{R} \to \mathbb{R} [/math] by [math]f(x)=x^{2}[/math] not being uniform continuous. Starting out with the definition:

[eqn]\exists \hspace{0.1cm} \epsilon > 0 \hspace{0.1cm} \text{such that} \hspace{0.1cm} \hspace{0.1cm} \forall \hspace{0.1cm} \delta > 0 \hspace{0.1cm} \exists \hspace{0.1cm} x,y \in \mathbb{R}, |x-y|<\delta \implies |f(x)-f(y)| \geq \epsilon [/eqn]

the proof starts with: let [math]\epsilon = 1[/math] and write [math]x = y + \frac{\delta}{2}[/math]. Why do we know that such a substitution will make this work and what is the process for coming up with such a number? Surely it's not glorified guess work.