Uniform Continuity...?

Question

Can somebody please explain to me what the heck is the difference between continuity and uniform continuity?

Answer 1

uniform continuity implies continuity, but not conversely. The first link seems to help quit a bit, the wikipedia not so much so, though it does have a good example with the details missing.


http://www.maths.nott.ac.uk/personal/jff/G12RAN/pdf/Uniform.pdf#search='uniform%20continuity'

http://en.wikipedia.org/wiki/Uniform_continuity

Answer 2

Nof47d78613943de44cc0c08ec384575e that's no longer sufficient. A function this is non-supply up on a closedf47d78613943de44cc0c08ec384575e bounded era is uniformly non-supply up on that intervalf47d78613943de44cc0c08ec384575e however the comparable isn't genuine for the entire actual line (that's closed yet no longer bounded). Considerf47d78613943de44cc0c08ec384575e for examplef47d78613943de44cc0c08ec384575e f(x) = sin (e^x). it somewhat is non-supply up everywhere and bounded via [f47d78613943de44cc0c08ec384575ef47d78613943de44cc0c08ec384575ef47d78613943de44cc0c08ec384575ef47d78613943de44cc0c08ec384575e]-a million,a million yet no longer uniformly non-supply up on R on account that as x gets largerf47d78613943de44cc0c08ec384575e further and further oscillations are squeezed into the comparable area; in case you p.c.. a given ? you could consistently detect a ok such that between ok and ok+? there is an entire cycle of the sin function in between them. i think of the best thank you to flow approximately proving uniform continuity on your case may be to seem at f(x+?) and notice in case you could derive an bigger certain for |f(x+?) -a million,a million f(x)| in terms of |?| on my own (i.e. no longer based on x). in case you're able to do thatf47d78613943de44cc0c08ec384575e you will desire to be able to tutor uniform continuity; for a given ?f47d78613943de44cc0c08ec384575e you only would desire to locate a ? that makes your bigger certain decrease than ?.

Answer 3

I think, one goes in circles and the other one is squared.

Answer 4

They beat me to it.

...but it's good to re-visit the subject ..thanks