for (1),example: .
for(2), take f(x) as above mentioned, and .
1) Assuming f is a real-valued function defined on [a,b] of the real line, f is differentiable on the whole [a,b], is it possible that does not exist (both finite and infinite) while exists?
2) Assuming function , E a subset of real line, let E' be the subset of E on which is defined, is it possible that E' contains isolated point?
Thank you for the first example. Because the formula can not be displayed, I rewrite it as follows: Define , then f'(0)=0 so f is differentiable on the whole [0,1], but does not exist (discontinuity of the second kind).
As for your answer to my second question, I think I've got it although some correction may be needed. The domain E should be . 0 is the limit point of E, so any limit of function remains the same and f is differentiable at 0 as a result. But sorry for my vague expression in the orignial question, I mean that the isolated point in E' is with regard to E, that is, the isolated point of E, not merely of . This is not the case for this example because 0 is a limit point, so not a isolated point, of E.
For clarity I restate the second question as follows:
2) Assuming function , E a subset of real line, let E' be the subset of E on which is defined, is it possible that E' contains element which is an isolated point of E?
Could you please help me with this question?