Every Function with a Bounded Derivative is Uniformly Continuous Proof

In this video I prove that every function with a bounded derivative is uniformly continuous. I hope this video helps someone out there who is studying mathematical analysis/advanced calculus.
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Thank you:)

[@jazmineesparza4152]

Awesome. I've been drowning in my class and was praying you'd upload more real analysis stuff. Your videos are seriously the best out there for this course, thank you!!

[@TheMathSorcerer]

aww thank you! So happy it is helping:)

[@davidj2452]

Its so fun to see your thought process! I am taking a class on nonlinear control systems and I never had any background in real analysis or proofs. So seeing how you go through them step by step is a tremendous help!

[@jake_lizzard9303]

Hey fantastic video! I would like to see more videos where you try to prove things on the spot and take us through your thought process like in this video. Have a nice day!

[@TheMathSorcerer]

Will do lots more coming!!

[@mgarner7439]

Thank you for these videos! You are seriously saving my sanity

[@TheMathSorcerer]

You are welcome, I'm so happy this is actually helping someone:)

[@learngrowentertain2510]

Very Well explained respected Sir. Great explanation...

[@maxpercer7119]

Uniform continuity is a hurdle I never completely got over.

[@TheMathSorcerer]

Yes😄

[@letpieau1660]

Thank you for your video. I am wondering do we need to show f is closed on a closed interval to use MVT, since it is not given.

[@patrickmederitsch1380]

If you assume the domain of f to be the reals, then it follows that f is continuous on the reals, since we know that f is differentiable (this is something you might need to prove). But when f is continuous on the reals, it is for sure continuous on any closed interval, since this closed interval must be contained in the reals. Hope that answers your question.

[@ratulchoudhury9144]

Well explained

[@DineshSingh-qw8oh]

Thank you Sir 🤗

[@TheMathSorcerer]

You are welcome!

[@NirodhaLL]

Just when i thought i was done with uniform continuity, great video!

[@TheMathSorcerer]

Thanks man!

[@chanukyachowdary1252]

Sir, Is this an if and only if conditions for uniform continuity?

[@N7_CommanderShepard]

This is why I’m a physics major. I don’t understand proofs haha.

[@TheMathSorcerer]

Haha well physics is hard I think!!!

[@jakepete]

i'm still unsure why you can use MVT without first proving that it's continuous?

[@Aman_iitbh]

derivative exist means its continous

[@tanish6035]

Infact, f becomes lipschitz and hence uniformly continuous.

[@Aman_iitbh]

lipschitz isnt imply diffrentiablity , consider |x| its lipschitz (consider reverse triangle inequality) .u mean to say uniformly cont

[@tanish6035]

@@Aman_iitbh oh yaa, i just wrote it mistakenly, thanks for correction.

[@HitheshAvishka0215]

You look like JEFF BEZZOS !!

[@oscardavidalarcon2673]

Yeah lml

[@TheMathSorcerer]

😄