Prove that

does not exist.

Proof:

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

.

***** If a limit L exists, then f(x) approaches that limit, hence f(x)-L must be at least <1 at some point*******

We consider two cases.

Case 1.

. Consider

.

******But if

x must be

******

Then

. However,

. So

, a contradiction.

******

********

Case 2.

. 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.