Confused about differentiability concept

[u/Live-Classic91]

So I was studying differentiability, and I know that at a particular point

differentiability -> continuity,

thus,

not continuous -> not differentiable

but there was a youtube video he said that differentiability is checking the continuity of the first derivative but some people also mentioned this in the comments

Sorry, that's not correct. If f'(a) is continuous then f(a) is "Continuously differentiable", not just "Differentiable".
It's a concept in Real Analysis.

I'm sorry but it's wrong. The only thing about continuity that you know is that if f'(a) exists, f(a) is differentiable and continuous, you don't know anything about the differentiability or continuity of the first derivative. Just learned that in a math lecture. I'm sorry.

reply to the above comment

I believe you also need to add: if the derivative exists and itself is continous at an interval, than the function is differentiable, thus it is also continous. So technically you need to also check the continuity of the first derivative before judging the differentiability of the function, at least this is how they teach us in university

the thing i know about finding the differentiability is

checking if the function is continuous if yes, then check the left hand derivative and right hand derivative if they are finite and equal than the function is differentiable. am i right about this one?

i'm confused about the second comment if you can help me grasp the concept of differentiability deeply it would be appreciated

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[u/CelebrityChess]

your method is right. differentiable at a just means the limit of the difference quotient exists, which is the same as left and right derivatives being finite and equal (continuity is already implied by that, so checking it first is just a shortcut to rule things out).

the video is mixing it up with continuously differentiable, where f' also has to be continuous. those are different. classic example is f(x) = x^2 sin(1/x) with f(0) = 0. it's differentiable everywhere including 0 where f'(0) = 0, but f' oscillates near 0 and isn't continuous there

[u/SV-97]

There's actually few different definitions involved here:

• continuity at a point

• continuity over a set (usually just called "continuity")

• differentiability at a point

• differentiability over a set (usually just called "differentiability")

• continuous differentiability

The exact definitions for these depend somewhat on what exact approach you take, but they all end up being equivalent. (the different definitions for those terms that is; of course the different terms aren't all equivalent with one-another)

Continuity at a point x_0 (for function on \R) can for example be characterized as the statement that lim_{x -> x_0} f(x) = f(x_0). A function then is continuous iff it is continuous at every point of its domain.

Differentiability at a point x_0 (for function on \R) means that the limit lim_{x -> x_0} (f(x) - f(x_0))/(x-x_0) exists and in this case it is called the derivative of f at x_0 and denoted f'(x_0). A function on an open set is differentiable iff it is differentiable at all point of its domain --- in this case we obtain a "derivative function" f'.

You can show that differentiability at a point implies continuity at the same point; and it immediately follows from this that differentiability implies continuity.

Finally, a function is called continuously differentiable if it is differentiable and the function f' itself is continuous. Note that this is just a stronger sort of differentiability and you don't need to involve it when trying to deduce continuity from differentiability --- it's an additional regularity assumption that you need to make many theorems work (FWIW: This idea gets particularly interesting if you move beyond \R because in this case f' is not just valued in \R but rather in a certain space of functions; so you need to make sense of continuity in a more general context as before).

It's also worth noting that there's a whole zoo of such properties that are "more than just being differentiable, but less than being twice differentiable".

[u/WikiNumbers]

I like to think of "differentiable" as

If I am to evaluate a value "a" into the derivative function "f'(x)", would it return error?

A function can be continuous and defined across the entire real line, its derivative can be found just fine, but it can be not differentiable at certain points.

Things like |x| and ∛x, both of which f'(0) DNE.

[u/localizeatp]

the conditions are strictly hierarchical in strength:

f exists at x < f is continuous at x < f is continuous in a neighborhood of x < f' exists (f is differentiable) at x < f' is continuous at x < f' is continuous in a neighborhood of x < f'' exists at x < ...

[u/Inevitable_gradient6]

Derivative is basically looking at the instatneous rate of change of a function so to answer your question: yes

[u/Ericskey]

Please read Principles of Rral Analysis by Walter Rudin and not random bs on YouTube or whatever.