Let me introduce you the problem killing me.
Assume that and consider the following equation
for , where .
Prove: implies .
Note: We do not know whether y is differentiable or not.
The first method comes into my mind is proving by contradiction.
Suppose the contrary that but for some values in .
Then we may pick satisfying on and on for some .
For simplicity, set and .
Then, for all , we have
If , then letting with , we get and thus .
This is a contradiction.
But my method fails to deliver the case .
I need help at this point.
Your help is highly appreciated!