Continuous and Uniformly Continuous Functions

We outline the difference between "point-wise" continuous functions and uniformly continuous functions. Basically, with "normal" or "point-wise" continuity, for any given point, for every ε, we can find a δ such that if the points are within δ, the imges are within ε. However, with uniform continuity, given some ε, we have to find a δ that works for -every- point of the curve. In other words, we pick the point first for normal continuity, and we pick the points after we pick ε and δ for uniform continuity. Another way of thinking about this is that point-wise continuity is a "local" condition , while uniform continuity is a "global" condition.

I try to illustrate this the best I can through examples and visual demonstration.

[pseudorealityisreal]

Whaaaat!!!!??? 5 minutes is all it took for you to clarify a concept I was trying to figure out for months 😂...Excellent!

[JTehAnonymous]

That explanation was so clear. thank you very much.

[dongookson]

Man the last 20 seconds...cannot thank you enough. God’s work.

[annikabrundyn]

Amazing! Thank you! There aren't enough good videos on Real Analysis :)

[dk]

Its CRAZY how someone on yt can explain a concept much more efficiently. I am a math major in the first semester and I am rlly struggeling to understand concepts when professors explain it, or its just hatd for me to understand stuff in the lecture, even worse when friends try to explain it to me… as they are trying to confidently teach me, (the themselves haven’t understood it good enough) I then feel very stupid.
But I know it mostly depends on their explanation… so thank you!!!

[pankajaggrawal]

Great animation and explanation. It is first time, when I could understand uniform continuity geometrically.

[Whitecroc]

Thank you. This has bothered me for years. The definition is so abstract and features so many moving parts I was never quite sure if I got it.

[danielyang]

The graphical explanation cleared up all confusion I had about the definitions. Thank you.

[Stuks]

I struggled for 1 week trying to understand continuity. Now finally thanks to you i understood!

[viveakkatochG]

Hands down the best explanation on this topic . 💯🕺

[magno]

For the uniformly continuous counter-example, it would be nicer if you kept both the epsilon and delta fixed and moved the blue region closer to the y-axis and picked two points on the curve that are in the blue region but are clearly not entirely in the red region.

[znhait]

This is the best explanation I have seen explaining the difference between continuity and uniform continuity.  Unfortunately, the main thing with these problems is how difficult they are to actually prove.

[icee]

You saved me a weeks worth of frustration my friend. Bless you!

[noahz.]

Great examples and visuals. Very concise and no rambling

[Ke_eK]

Before finishing wathcing this video I didn't believe that this short 5 min video could actually help me but I was soooo wrong. Thank you so much.

[farukahmed]

This video has makes my understanding better of continuity. Very good video that's makes everyone impressed.

Thank you. You helped me. Nice explanation and nice visuals. :)

[michaelkisumu]

Awesome Video! Very clear explanation of the use of Delta-Epsilon in the context of uniform continuity, and the counter example added even more clarity

[kiwanoish]

Very nice video, and a very clear and concise explanation! Note though: Starting at 2:50 it says at the top right: "If a given \delta works for any \epsilon we choose, for any points in the domain". This might be a slightly confusing formulation, since it's more like: For a given \epsilon, we can find a \delta that works for any points in the domain. We cannot find a \delta that works for any \epsilon. But, again, you explained it perfectly, and hopefully people who watch the video will get the right idea anyways. Thank you!

[AlpstoonE]

Great explenation! Really helpful for my upcoming calculus exam!