This is a fascinating question about the transcendence properties of the normal distribution function. Let me analyze this carefully.

The question asks whether for rational numbers implies that necessarily and .

First, let's verify that is indeed a solution:

This follows from the symmetry of the normal distribution around 0.

Now, the key question is: are there any other rational points?

The answer is that we don't know for certain, but it is widely conjectured that there are no other rational points.

Here's what we do know:

1. The normal CDF is transcendental: The function can be expressed in terms of the error function:

   The error function is known to be transcendental (not algebraic).

2. Known transcendence results: For any non-zero algebraic number , the value is transcendental (Lindemann-Weierstrass theorem). However, this doesn't directly apply to integrals of .

3. Special values: We know that:

   • (rational)
   • (rational)
   • (rational)

   But these limits aren't achieved at finite rational points (except for ).

4. Related conjectures: It is conjectured that for any non-zero algebraic number , the value is transcendental. This would imply that is transcendental for any non-zero rational .

5. Current state of knowledge: As far as I know, there is no proof that is the only rational point on the graph of . This remains an open problem in transcendental number theory.

Conclusion: While it is strongly believed that is the unique solution with both values rational, this has not been proven. The question touches on deep issues in transcendental number theory that remain unresolved.