I am a little shaky with the concept of proving uniform continuity vs regular continuity. Is the difference when proving through epsilon-delta definition just that your delta can not depend on "a" (thus be defined in terms of "a") (when |x-a|<delta) for uniform continuity? Also to the more specific questions: if you have a function that is uniformly continuous for all rational numbers how can you prove that an extension of that function is uniformly continuous on all real numbers. I was thinking that from density of rationals and irrationals I can say that a rational number is necessarily flanked by two irrationals. Thus if we treat the rational number as an open interval between those two irrationals (interval which happens to contain only one point) then necessarily the function on the closed interval including them is also uniformly continuous. But I feel like that doesn't prove it for ALL irrationals - since technically between two rationals there's an infinite number of irrationals right? Am i totally off track here?