A community for students.
Here's the question you clicked on:
 0 viewing
gavyap
 3 years ago
How to proof problem set 1 question 1D10: Show that
g(h) = [f(a + h) − f(a)]/h has a removable discontinuity at h = 0 and that f'(a) exists.
gavyap
 3 years ago
How to proof problem set 1 question 1D10: Show that g(h) = [f(a + h) − f(a)]/h has a removable discontinuity at h = 0 and that f'(a) exists.

This Question is Closed

FabianMontescu
 3 years ago
Best ResponseYou've already chosen the best response.0The key word is "removable". Let's do this. The function g(h) is discontinuous when h = 0. The limit of g(h) when h goes to zero is f'(a). For the right arrow: We know that the function has a removable discontinuity at h = 0. By the definition of removable discontinuity, the limits when h approaches zero from the left and from the right are the same. That is also the definition of a differentiable function with the formula g(h), and that proves the right arrow. For the left arrow: If f'(a) exists, we know that the limit of g(h) when h goes to 0 from the left is the same as the limit of g(h) when h goes to 0 from the right. That's the definition of differentiability: the right and left limits of f'(a) = lim h>0 g(h) are the same. We also know that g(h) is discontinuous when h = 0, but because its limits from the left and from the right when h approaches zero are the same, we know that the discontinuity is removable. This proves the left arrow.
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.