Prove that does not exist.
Let Assume, to the contrary, that exists. Then there exists a real number such that . Let . Then there exists such that if is a real number satisfying , then . We consider two cases.
Case 1. . Consider . Then . However, . So , a contradiction.
Case 1. . Consider . Then . Also, . So , a contradiction.
Remarks: I am particularly troubled by the choice of 's being contrary to the values of . Intuitively, we know that the value of cannot be negative when and cannot be positive when .
Please tell me where I am wrong.