I'm seeking a bit of affirmation or correction here before i try to solidify this to memory....(adsbygoogle = window.adsbygoogle || []).push({});

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 .

Iinterpretthis 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).

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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.

Iinterpretthis 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.

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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!

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# Continuity & Uniform Continuity

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