Question 1

I'm just wondering, at 7:08 why did Sal choose the lim x->c(f(x)-f(c))? Was it just an arbitrary limit that he chose?
Thanks

Accepted Answer

He chose it because someone else worked out that this procedure gave the desired proof ...

Doing proofs involves a lot of trial and error!

Question 2

HELP! I've been thinking about this topic for a little bit. I've watched the video a couple of times and consulted a few other sources, and I can say that I am thoroughly confused.

How exactly does this proof show that Differentiability implies continuity? I can follow the math. But, I think that I'm missing something here. What are the key parts of this proof and how does it link together the topics of differentiability and continuity?

Accepted Answer

This proves that differentiability implies continuity when we look at the equation Sal arrives to at 8:11. If the derivative does not exist, then you end up multiplying 0 by some undefined, which is nonsensical. If the derivative *does* exist though, we end up multiplying a 0 by f'(c), which allows us to carry on with the proof.

Question 3

Just curious to know how others interpret ? 
Intuitively why limit x -> c (f(x) - f(c)) = 0. I understand the math but want to know what does that mean, and why it becomes 0 intuitively ? Thanks.

Accepted Answer

You're basically subtracting one number from itself  the limit of f(x) as x -> c means eventually f(x) will equal f(c).  so that means the subtraction problem becomes f(c) - f(c)

Question 4

By using "implies" does this mean that where a function is differentiable the function _*must*_ be continuous or does that mean that it _*strongly indicates*_ that the function is also continuous? (assuming the _must_)

Accepted Answer

Differentiability at a point *confirms* that a function must be continuous at that point. However, vice versa isn't true (You can have a continuous function with a sharp turn)

Question 5

At 10:11 Why did we assume that `lim x -> c f(c) = f(c)` ?

What if function f(x) is undefined at c or if `lim x -> c f(c)` does not exist ?

Accepted Answer

"What if function f(x) is undefined at c?": We assumed that f(x) is differentiable at x=c. This alone means f(x) cannot be undefined at x=c. 
And as tyersome answered, f(c) doesn't have an x, so lim x->c can be removed from "lim x -> c f(c)".

Question 6

at 2:51  is the sentence  "the function f(x) is differentiable at x=c... really just saying this llimit right over here actually exists" is it indicates that if the function f(x) is differentiable at x=c, then the limit exists and if the limit exists, then the function f(x) is differentiable at x=c, which is known as biconditional statement(iff)?

Accepted Answer

It is not bi-directional, 
In case if f(x) is differentiable at x=c, then limit exist, yes and and f(x) is continuous at x(proved in the above theorom).
And in case if f(x) is said to be continuous, we can't simply say it is differentiable, since its not just limit of f(x) we are calculating but limit of slope of f(x) is what we need to calculate to find differentiability property of f(x).
Hope this helpful.

Question 7

@3:24 definition of continuity : lim x->c f(x) = f(c)
If this holds; lim x->c f(x) - f(c) = 0, then shouldnt the numerator of differentiation quotient always be zero at point c?

lim x->c f(x) - f(c) / x-c

So then the derivative would be zero at that point algebraicaly but we know it doesnt have to be? Please comment what I'm interpreting wrong here ?

Accepted Answer

The differentiation limit, lim x->c (f(x)-f(c))/(x-c), will always be 0/0. However, 0/0 is indeterminate and can equal all real numbers as well as +/- infinity. What 0/0 equals depends on the specific function we are differentiating.

Question 8

I am trying to understand what “Differentiability Implies Continuity” means. I made the incorrect assumption that to prove differentiability, all I needed to do was confirm that:
        lim x->c- (f(x) – f(c))/(x – c) = lim x->c+ (f(x) – f(c))/(x – c)
The assumption was incorrect. All the above indicates is that the slope as ‘x’ approaches ‘c’ from the left and right is the same. It does not indicate that the function is continuous at ‘c’, i.e. lim x->c f(x) = f(c).  It seems that to prove differentiability, it is necessary to first prove continuity; which seems counter to the statement that “Differentiability Implies Continuity”. What am I missing?

Accepted Answer

I think your assumption, "lim x->c- (f(x) – f(c))/(x – c) = lim x->c+ (f(x) – f(c))/(x – c)" implies that "continuity implies differentiability".  However, Khan showed examples of how there are continuous functions which have points that are not differentiable. For example, f(x)=absolute value(x) is continuous at the point x=0 but it is NOT differentiable there. In addition, a function is NOT differentiable if the function is NOT continuous. In this video, Khan is merely proving that if you know the function is differentiable, then it MUST also be continuous for all the points at which it is differentiable.

AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 2

Proof: Differentiability implies continuity

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