Suppose that

is continuous, g(0) = g(1) = 0, and for any

there is some

such that:

and

.

Prove that g(x) = 0 for any x in [0,1].

My work:

Since g is a continuous mapping of a closed interval, it must achieve its maximum, say M, on [0,1]. So, let A = {

}. Since g achieves its maximum, A is nonempty. A is clearly bounded above by 1, so

exists.

Suppose that

. Take

. Then,

(since g(0) = 0 = g(1)).

Then, there is a k > 0 such that

and

and that

.

But, since M is the maximum of g, this implies that

.

This is where I'm getting stuck. Can anyone point me in the right direction?