i love this channel! Who says summer has to be enjoyed away from school? Not me! Math is the best!

I like the fact that it makes so much sense when you are also thinking on the derivative as a linear aproximation, you get the aproximation precisely in the same way, and basically when there is only one posible outcome for that aproximation, you have differetiability.

You might like to know about the Weierstrass Function, a function which is continuous _everywhere_ but differentiable _nowhere._

6:38 Actually you are using differentiable at this point right now, because limit of product is equal to product of limit if both limit exists. For example lim sinx/x cannot be decomposed to lim sinx lim 1/x, just to make it rigorous.

Thank you so much! Your explanation was so clear and concise, and your enthusiasm made it so enjoyable to watch! Thanks for being my quarantine lecturer!

I'm loving the high energy thumbnails!

There exist an alternative definition of differentiable function: Let f:[a,b]→R, then f is differentiable in Xo if exist a function g:[a,b]→R continuous in Xo so that: f(x) = f(Xo) + (x-Xo) g(x), and g(Xo)= f'(xo). By that definition f is continuous in Xo because it's a sum of continuous functions: f(Xo) and Xo are constant numbers, x is continuous in every x and g(x) is continuous by definition. #YAY

Beautifully done. You should make a college calculus course as you are excellent at going explaining your steps

The limit of a sum is not always the sum of the limit right? For example lim n->inf of ln(n)-harmonic sum till (1/n) converges. If you split up the sum into 2 limits, you get infinity-infinity, which is undefined. I think that you need to place the restriction that f(a) is defined. P.S. I know that if f(a) is not defined then the function is definitely not continuous at a, but still, you cannot split limits in general. Peace

Thank you, best video on YT on this proof!

The limit of a sum, is the sum of the limits. The same is true for the limit of products. My understanding is that this is true only if the functions inside the limits are continuous. (No?) But, we are trying to prove continuity in the first place. Can we prove continuity by assuming continuity first? It is clear that I am missing something important. Can you please clarify. I need this little clarification because I haven't studied calculus for almost thirty years and I don't remember much of it. (Love your channel, BTW. It brings back much of the love for math I had as a young person. So, thanks and keep up the good work.)

I suddenly find myself very curious about how to find the area of a fractal, as in the integral of a continuous function that is not differentiable anywhere.

Awesome proof, it finally made sense to me :) I know abs(x) at x=0 is the counter-example but can you also show in general that "if continuous then differentable" is a false statement or it doesn't lead to any conclusions?

This proof was taught to me in school. So I won't watch till the end, but I still liked and commented :) beautiful proof

man this shows why i never got proofs. whenever i read them i tend to forget what i wanted to show in the first place or what the proof is supposed to show^^

Can you explain why f'=0 implies f is constant if the domain is the reals, but not if the domain is the p-adic numbers?

The way I'd prove this is by saying: "If you take the derivative of a function in a point *a*, you will have the slope the line tangent to the function in that point, but you can only have the slope of line tangent the function at that point if, and only if, the point *a* exists in the function (if it is continuous at a), since you can't have the slope of nothing" For example we have the famous function sin(x)/x, we can clearly see it's not conitnuous at 0, so that point doesn't exist in the function. It's derivative is (xcos(x)-sin(x))/x^2 , and you can see that the slope of the tangent at 0 is 0/0. sin(x)/x it's not differentiable at 0 because the point 0 doesn't exist in that function! :3 Is my proof right? QwQ

Is it rigorous to say that the product of a derivative and an infinitesimal quantity is always 0?

hm.. |0| = 0. But for f(x)=|x| df/dx at 0 don’t exist?

Does this count as a real analysis question?

Clickbait: Table → Content. 😁

Can you help me with the following integral? Integration of x^2 * √(1+4x^2) dx It seems to me you can use integration by parts or something like trig substitution. However i get different results when doing so

integration of 1/x^2 + 3x ?

Great video. Thanks again.

Great video as always

Hello sir can I have a favor? do you lecture notes for Differentiability in R^2?

thank ypu so much

Oh man this came up in my analysis exam two months ago :0

You are amaizing thank you

What was that ending? Vihart is taking over your channel?

Wait at the end it looks like the tangent formula

Isn't a differentiable function by definition continuous?

When you record a snail you find in your backyard

Can you solve this please? X^lg(X-2)=1000

What happened to the video where you found the limit of infinite square roots of 0? #yay

Nice

Will you discuss how the converse is not true and mention the Weierstrass function? en.wikipedia.org/wiki/Weierstrass_function

"I don't know what a number is..." Are you sure you are the right person to make math videos?

is there any other example of continuos functions that are not differentiable at some ponit?

Lab glasses??

#yay!❤

I want a tshirt that says d'able

Sir, I don't get the proof..you just proved lim x → a f(x)=f(a)....where is the differentiability here?.. we are asked to prove if a f is differentiabie it is continous

8:44 it’s me on math classes))))))

Does continuity also imply differentability?

#yayyyyy

You should've put another minute or two of that snail to get to 10 minutes. Do you not watch PewDiePie?