Doubts from function

I need solution of this

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A putnam problem

What if f(x)=x

F(x) = x doesn't satisfy the condition f(x) >_ 0 for all x belongs to R

Sorry for wrong thought

This problem solved in stackexchange seen 2 or 3 month before

Can it be done like this ?

I don't think that this give contradiction

A example

Bro this function is not infinitely differentiable, infact any polynomial with finite degree will be finitely differentiable

No bro that function is infinitely diffentiable

That promises continuity of function , not infinte differentiability

Any polynomial is infinitely differentiable

Since f:R-R is under consideration ,so we can take 'I' to be the real number line . Hence we can assume f to be a function infinitely differentiable over all x€R

Just an example,we can solve graphically too ig