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# Proof: Differentiability implies continuity

Sal shows that if a function is differentiable at a point, it is also continuous at that point.

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• Does the opposite also imply ie. continiuity implies differentiability ? if so why and why not ?
• When Sal starts his proof of how differentiability implies continuity, why does he only use the numerator (f(x)-f(c)) and drops the denominator (x-c)??
• This is a common way of making demonstrations in mathematics: you start from an expression A, and by simplifying it you show that it is equal to an expression B. Then starting from A again, you show that it also simplifies to an expression C. You can then say that B = C. The whole point is that, by starting from A, you only make simplifications, but if you had to start from another expression, you would have to go through more complicated steps to reach A.
• Everything is very clear except for the part where we "decide to take a look at a slightly different limit " which is lim { f(x) - f(c) }. How do we decide to take a look at it... it seems like it is "planned" (so that later we just multiply it and divide it by (x-c). If you understand what I am trying to say. Or is it assumed at that point that the lim { f(x) - f(c) } is the same as if we wrote lim f(x) = f(c) which is the definition of continuity ?
• We indeed derive lim x→c { f(x) - f(c) } from lim x→c f(x) = f(c). Because f(c) is a constant, after removing it from both sides we can put it inside the limit.
lim x→c f(x) = f(c)
[lim x→c f(x)] - f(c) = 0
lim x→c {f(x) - f(c)} = 0

Edit (see comment): This is not a proof but an explanation for the derivation of the expression which will in turn proove that differentiablity implies continuity.
• Why did he use limit x>c f(x)-f(c)?
• At Sal writes defines continuity as lim x→c f(x) = f(c).
He then uses lim x→c f(x)-f(c) and shows this equals zero. Let's see what this gives us:
lim x→c f(x) - f(c) = 0
[lim x→c f(x)] - [lim x→c f(c)] = 0
lim x→c f(x) = lim x→c f(c)
Now, the right-hand side is just f(c) because it doesn't have x in it. So we've got:
lim x→c f(x) = f(c)
which is our definition of continuity, which we wanted to show.
• When Sal says a limit "equals" something, that is a simplification or a less formal way of stating what is happening correct? If I am not mistaken, I thought limits only "tend toward" or "go to" a value, but never actually reaching that value all the way.
• The limit is a fixed value. Like 1. A stationary value or object cannot approach anything, but it can be approached. X is the value that is changing. We plug something in, and it will work along that function AS LONG AS IT IS NOT THE LIMIT. It can get very close, of course.
Think of the way we say the notation, "What is the LIMIT of f(x) as X approaches 'some value?'".
• I understand the proof but what about this example? Let's say a car is moving along the road, time is independent variable and distance is dependent. Then velocity at some moment of time has to be the derivative. At some moment car hits a wall, crashes into it and loses all the velocity (completely ineslastic collision). At the very last moment the car had some velocity (differentiable) but suddenly lost it afterwards (not continuous). Where's the mistake?
• On a small enough scale, the velocity of each particle of the car will experience an electromagnetic force with the particles in the wall. Provided that the wall is sturdy enough to handle the collision, each individual particle will feel an abrupt acceleration forcing them to slow down and recoil. This is all one continuous motion, but it takes place on a small scale of both position and time.
• I wonder in the first graph where f(c) is a point other than on the curve, here f(x) is defferenciable at c, that means limf(x) - f(c) =0, which is saying f(x) is continuous at c, obviously it's not right, how can explain that?
• If f(c) is not on the curve of the function then that is a discontinuity and the function is not differentiable at that point.
• : Is assumption correct ?

We assumed that `lim (x > c) f(c) = f(c)`

We can only assume this when we already know that f is continuous at c.

But the whole point of this proof was to show differentiability implies continuity.

Am I missing something here ?
• In order for 𝑓(𝑥) to be differentiable at 𝑥 = 𝑐 the function must first of all be defined for 𝑥 = 𝑐, and since differentiability is a prerequisite for the proof we thereby know that 𝑓(𝑐) is indeed a constant, and so
lim(𝑥 → 𝑐) 𝑓(𝑐) = 𝑓(𝑐)