The notation:

\lim_{x\rightarrow c} f(x)=L

... means:

∀ε>0:∃δ(ε,x)>0:∀x:(0<|x-c|<δ\Rightarrow |f(x)-L|<ε)

Continuity

\lim_{x\rightarrow c} f(x)=f(c)

Uniform Continuity

If δ is only a function of ε, not of x.