Here is attempt #2:

Let's momentarily assume that the conclusion is correct.

Consider:

Now, by induction, we can try to show that

for any positive integer n, and any real number,x. We showed the base case above. Now, the induction step...

This shows that if we have a positive integer n, then:

Then, we show that it works for negatives...

So we have that,

Now we try to show that

works too...

. So we have that

We have shown that that for integers in

:

Using our results above we can look at rationals...

So we have shown that when

is rational,

Now, we have to show that this works for all real numbers...

First, suppose we have any real number x. Then, there exists a sequence of rationals, lets call it q(n), converging to that x. Above we showed that

However, since

is continuous, this means that

. Therefore, for any

:

I think this is a better proof??? Although I know I kinda assumed the conclusion at the beginning. Any more feedback would be appreciated.