Question

I have a question in the problem set one: 1D-9
here is it : prove the following g(h)=(f(h+a)-f(h))/h, and the function g(h) has a removable discontinuity point at h=0; so we can conclude that f(a)' should exit.
thx inadvance

Answer 1

Well, by definition:
\[g(h) \simeq f'(a) = \frac{f(a+h)-f(a)}{h} \\ h\rightarrow 0 \] When h = 0 exactly, the function is undefined, not continuous, yet if f'(a) exists, it must be removable because \[ f'(a +h) = f'(a-h) \] as h approaches zero.

Answer 2

thx datanewb

Answer 3

No prob.