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uncut fableBOT
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untold skiff
Well, what's the definition of continuity?
Okay. By definition, a function f is continuous at a point a if and only if lim_(x -> a) f(x) = f(a).
I mean, can you prove iii?
...no, I mean... proof.
Well, let's start with this; what does it mean if f is differentiable at a?
What is f'(x)?
Which is defined how?
...no, it's defined as a limit.
Yes.
Except we want f'(a), so we can actually change it a bit.
Change all x's to as, then let x = a + h.
Then h = x - a.
And as h approaches 0, x approaches a.
So we get $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$
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untold skiff
Now, if f is differentiable at a, this limit must converge, right?
...a convergent limit is one that equals a finite value.
You really should know this stuff.
It means it doesn't go to infinity and it doesn't bounce between two or more values.
...I don't know how that's a response.
Which means there exists some $L$ such that $\lim_{x \to a} \frac{f(x) - f(a)}{x - a} = L$
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untold skiff
no.
No.
What do all of those theorems have in common?
Do we have an interval here?
...no.
There isn't one.
Okay, stop.
Calm down.
The point is that none of those theorems about intervals apply.
Because we don't have an interval.
We're just looking at this limit.
Please stop trying to guess. Guessing is bad in math. There are literally infinitely many answers and exactly one of them is correct. It is statistically impossible to be right by guessing.
I'm not trying to lecture, just to educate.
Okay, so.
$\lim_{x \to a} \frac{f(x) - f(a)}{x - a} = L$
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untold skiff
Which, after a bit of algebra, means $\lim_{x \to a} f(x) - f(a) = \lim_{x \to a} L(x - a)$, right?
static kayakBOT
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untold skiff
Now, what do you notice about the right side?
Remember, L is just a constant.
What is $\lim_{x \to a} x - a$?
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untold skiff
And does multiplying it by a constant change anything?
Right.
Therefore $\lim_{x \to a} f(x) - f(a) = 0$.
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untold skiff
Which means $\lim_{x \to a} f(x) = f(a)$.
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untold skiff
Which is precisely the definition of f being continuous at a.
...it was... a proof... that differentiability implies continuity.
Yes, but the important point is to understand why.
It is C, yes.
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