Suppose I have a compactly supported function $f$ defined with $supp f \subset I= ]0,1[$. Let $K=supp f$.
Is this statement true:
We can assume without loss of generality that there exists $ \alpha, \beta \in I$ such that $ K \subseteq [\alpha , \beta ]$ and $ f(\alpha)=f(\beta)=0$.
My idea how to start is so good that a start becomes a finish. :-) Put $$\alpha=\frac {\inf\operatorname{supp} f}2,\beta=1-\frac{1-\sup\operatorname{supp} f}2.$$