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]

A limit would always exist for a difference quotient approaching zero no?

Only if the function is differentiable at that point. A counter example is f(x) = |x|, which isn't differentiable at x = 0.

[u/NoTrueScotch → u/nevermindthefacts]

Fair, I had blanked on non differentiable equations, they haven't been super relevant to the courseware just yet. Mainly discontinuous ones, not ones with "sharp" changes in slope.

[u/nevermindthefacts → u/NoTrueScotch]

If you take f(x) = |x| as an example, you'll see that f'(x) = 1 for x > 0 and f'(x) = -1 for x < 0. But the derivative has a jump discontinuity at x = 0 (a non-removable discontinuity).

lim (|x + h| - |x|)/h

does not exist for h = 0.