Real Analysis 32 | Intermediate Value Theorem

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:14 Introducing the Intermediate Value Theorem
01:26 Definition Intermediate Value Theorem
02:29 Corollary
02:56 Proof of the Intermediate Value Theorem
08:16 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives
#Studying

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[lucaug]

One of my favorite theorems from Analysis, together with MVT :)

[kingarthr]

Funny that the "Intermediate value theorem" is the halfway point of the playlist

[Vanbaan]

Very cool theorem. Interesting to note that the opposite is not true. Let x in [a, b] and f be continious. Then f(x) is not necessarlly in [f(a), f(b)]. The proof pretty much uses the bissection method to find the root of f tilde.

[Independent_Man]

when you define f tilde in 4:29, shouldn't it be f tilde = -g when g(a) > g(b) and f tilde = g otherwise ?

Because at 3:58 you said that you wanted the value on the right to be larger than the value on the left.

[Hold_it]

Interesting Theorem. Keep them coming! :D

[tychovanderouderaa]

You define \tilde{f} to be -g if g(a) > 0. Then it is very clear that always \tilde{f}(a) ≤ 0, but to me, it is not directly obvious why also \tilde{f}(b)≥0. Let's say instead of the drawn function f, f is a parabola that is flipped at the minimum to get g. Wouldn't then both g(a) and g(b) be less or equal than zero?

Thanks in advance! I'm probably missing something obvious.

[Lous_-sthi]

Thanks for sharing such an insightful proof.

[aidynubingazhibov]

Are {an} and {bn} Cauchy sequences because these are monotonic and bounded (and thus convergent)? Just like in Bolzano-Weierstrass theorem? Also, what books are you using for the videos if any?

[someperson]

With this construction, what if \tilde{f} has multiple 0's? Couldn't you then miss the \tilde{y} you're looking for, and end up converging to a different 0?

Well actually now after writing this I suppose it's enough that there exist sequences which converge to \tilde{y}...

[fabiomendez-cordoba]

Hi, is there problem with the website? It keeps telling me that it can't be found :c

[meshachistifanus]

Proof, suggest a textbook to read along with.

[duckymomo]

A discontinuous function can have intermediate value property

It was a long debate whether IVT defined continuity and unfortunately we discovered that does doesn’t and in fact we get various levels of continuity (Lipschitz and absolutely continuity)

[johnstroughair]

We have shown [f(a), f(b)] is an interval but is this enough to show f[[a, b]] is an interval?