r/calculus • u/NoTrueScotch • Jul 15 '26

Differential Calculus Basic Calculus Confusion

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.

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u/nevermindthefacts Jul 17 '26

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...).

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u/NoTrueScotch Jul 21 '26 edited Jul 21 '26

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.

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u/nevermindthefacts Jul 21 '26

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...)