[Solved] Nonnegativity of y implies the same for x

Dear friends,

Let me introduce you the problem killing me. (Headbang)

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!