I'm seeking a bit of affirmation or correction here before i try to solidify this to memory.... I know continuity to mean: Let f:D -> R (D being an interval we know to be the domain, D) Let x_0 be a member of the domain, D. This implies that the function f is continuous at the point x_0 iff for any e >0 there exists a d>0 such that x belongs to the domain, D AND |x-x_0|< d => |f(x)-f(x_0)| < e . I interpret this to mean: This is the criterion by which we judge if some function (f) is continuous at whatever-point-we-wish-to-test-for-continuity-at (x_0) over some interval that is, in the least, a subset of the domain (if not the entire domain itself). //////// I know uniform continuity to mean: Let a compact set, K be a subset of R. Let f:K->R. Then f is uniformly continuous on the set K. I interpret this to mean: The previous definition of continuity is now applicable to any and every point that is a member of the compact set, K. In other words, the interval/set over which K is defined satisfies the previous criterion of continuity at all points in K. ==== Is there a need to adjust either my definition (as quoted by my prof. for an introductory advanced calculus class) or my interpretation of these concepts - or are they within a reasonable tolerance of "precise-ness" for the _actual_ definition/interpretation/distinction of the concept of continuity and of the concept of uniform continuity? Please advise, thank you!