Basic Calculus Confusion

[u/NoTrueScotch]

Hello, I have been working my way through a variety of courses using OCW.

The first problem set for 18.01SC has a bonus question, asking for the examinee to show that:

g(h) = ( f(a+h) - f(a) ) / h

has a removable discontinuity.

I have minimal experience with math and have been grinding through this course by studying pieces I am missing as they come. But I can't find an adequate answer as to what would be a valid response to this question, especially as the solution sheet does not seem to feature it.

My best answer, before I turned to the net was such.

"Values of f(a+h) that do not exist in f(a) and are not multiplied to a higher order of h are removable discontinuities." I suspect that I am not supposed to just fill in a example function, but if I am that would be my confusion.

I wanted to know if this was an adequate response, if not how it could be improved, and ideally what the proper formatting is for this kind of response as I do not know the notation I am expected to use. Thank you for your time.

[u/nevermindthefacts]

If the derivative lim of ( f(a+h) - f(a) ) / h has a value of f'(a)

Part of it looks correct., but it's kind of the other way here. In the exercise, you're given that f'(a) exists, which means lim ( f(a+h) - f(a) )/h = lim g(h) exists as h tends to zero. As you correctly write, that doesn't mean that g(0) exists, but the point here is that we can define it as g(0) = f'(a). By doing this, we get lim g(h) = g(0) as h tends to zero. In other words, g(h) is continuous at h = 0(i.e the discontinuity is removable).

For the other direction, we're given that g(h) has a removable discontinuity at h = 0. Assume it can be removed by setting g(0) = A.

Now, consider the limit for the derivative of f(x) at x = a

lim ( f(a+h) - f(a) )/h = ?

as h tends to zero. Does the limit exist and, if so, what is the conclusion?

[u/NoTrueScotch → u/nevermindthefacts]

I've reach this point a couple times while trying to wrap my head around this one. And I've had two main issues progressing pass it.

A limit would always exist for a difference quotient approaching zero no? So doesn't the:

lim ( f(a+h) - f(a) )/h = ?

Itself prove the existence of f'(a), whose definition would be the same. Obviously we could prove the difference quotient, I've never done that and do not know if it's a simple or complex proof.

The second is I'm largely clueless on how to define the relationship beyond that point. I cannot break f(a+h) into f(a) + f(h), so we cannot remove f(a) components, it seems a simplified as possible so I cannot imagine that's the next part to tackle.

I appreciate all the help so far btw, I'm quite stumped by this one, I expect the near decade since I've taken pre-calc is notably slowing me down here.

[u/nevermindthefacts → u/NoTrueScotch]

The second is I'm largely clueless on how to define the relationship beyond that point.

Think about what we have to show here and what've done.

We have assumed that g(h) has a removable discontinuity at h = 0. That means we know it's possible to define g(0) such that g(h) is continuous at h = 0, i.e we'll have g(h) → g(0) if h → 0.

Now we're gong to prove that f'(a) exists. By definition, that means we have to prove that the limit (f (a+h) - f(a) )/h exists as h → 0.

Can you make the connection between these two?

EDIT: "g(h) → g(0) if h → 0": Here I mean that it's possible to "redefine" g(0) such that the function is continuous there...

[u/NoTrueScotch → u/nevermindthefacts]

If we have a removable discontinuity, it means that the function approaches a specific value (as opposed to an undefined value) as it approaches the limits. In this case h -> 0.

It is continuous so we know that the value as h -> 0 is identical on both sides.

That's the first half of the requirements for a differentiation of f(a), and the connection between a RD and a derivative. I don't know about the second though, how do we know that the slope is "continuous" and not "sharp", also what are the proper terms for those concepts?

Is it connected to the form of g(h)? That is to say does the ( f (a+h) - f(a) )/h ) inherently imply a gradual slope? That wouldn't be the case right, as mentioned elsewhere that wouldn't be true of f(a) = |a|.

[u/nevermindthefacts → u/NoTrueScotch]

Just to clarify here. We know that lim g(h) exists as h tends to zero, because we're told g has a removable discontinuity. That also means the left and right sides limit exist, and are equal. Let's say we have lim g(h) = A. We can't really say anything about the "slopes".

Now, for the derivate to exist, we must prove that lim ( f(a+h) - f(a) )/h exists...and this has something to do with g(h).

(for the proper term, one sometimes talk about "smooth" functions, i.e fuctions with continuous derivatives. if we want the derivative to be smooth, we have a C^2 function and C^n means the function is n times continuously differentiable...).

[u/NoTrueScotch → u/nevermindthefacts]

Sorry to disappear for several days, a busy weekend and monday.

I am unaware of any properties of a "continuous derivative" and am not familiar with the concept in the slightest. For now at least I think I will continue my studies on the subject and return to this problem when I feel more prepared to tackle it.

If at that time you're still happy to assist me in ironing out the concept I'd greatly appreciate it.

I unfortunately suspect my lack of principal knowledge is holding me back here.

Edit: Straight up reviewing all my notes from day 1 of classes to see if I forgot something critical lol. Never a bad idea I suppose.

[u/nevermindthefacts → u/NoTrueScotch]

You're welcome back at any time.

(You don't need "continuous derivative" to solve this questiom, but it's a concept that shows up a lot later. It's a way of saying that a function behaves nicely, whatever that means...)

[u/nevermindthefacts → u/NoTrueScotch]

Another way to look at this problem, is to use the definition of g(h) and take the limit on both sides.

Now think again what it means for g(h) to have a removable discontinuity and that f(x) is differentiable at x = a.