as thus is not continuous at is of course, differentiable but not of class 1.
Let for and
a) Prove that is differentiable on .
b) Prove that is not continuous on .
I need help on BOTH parts. I've tried this problem, but I think I am not being very explicit in my proofs. I am also very sure my proof in b) is not nice. Is there another way to do b) [possibly without using the fundamental theorem of calculus]?
Attempt:
a) Clearly is differentiable at . [How do I show this?] In fact, [valid for ]. At , . Hence, is differentiable on .
b) . Let . Then because . So . So by FTC s.t. . So . But then is not a continuous function. But by FTC, is differentiable. Then , are differentiable functions whose derivatives are not continuous.
Thanks.
For part a), to justify that is differentiable on , you can say that it is obtained by product and composition of the following differentiable functions: on , on and on . The computation at 0 is fine (well, it would be better not to write when you don't know that it exists yet)
For part b), you don't need to refer to FTC : just say that tends to 0, while has no limit as , so that has no limit as . Therefore, it can't be continuous (whatever is).
No, is bounded near 0 (and it has no limit when ).