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).

////////

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.

====

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!

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Continuity & Uniform Continuity

Loading...

Similar Threads - Continuity Uniform Continuity | Date |
---|---|

I Understanding uniform continuity.... | May 9, 2016 |

Difference between continuity and uniform continuity | May 7, 2015 |

Local uniform continuity of a^q | Aug 19, 2014 |

Proof uniform convergence -> continuity: Why use hyperhyperreals? | Feb 15, 2014 |

About uniform continuity and derivative | Apr 5, 2013 |

**Physics Forums - The Fusion of Science and Community**