f is a continous function in (0,1) which maintains lim x-->0+ f(x)=-1, and lim x-->1^{- }f(x)=1

let's define: A={x in (0,1)| f(x)=0}

s=supA. i need to prove that f(s)=0.

well, i know that in oter words i need to prove that s is part of the group A, and is its maximum. But i don;t know how to proceed from that, because

my proffesor isn't satisfied with verbal proof. he wants formal mathematical proof [i saw him proving existence of sup using the definition of the limit (epsilon-delta) and by that contradicted the existence of another upper-bound number which smaller than the one we needed to prove]