Let I:=[a,b] and let f: be a continuous function on I such that for each x in I there exists y in I such that . Prove that there exists a point c in I such that f(c)=0
I think you can say something along the lines of take some f(x) in R, and w/ . We can find a such that . Suppose there was minimal value c, such that But we know we can find another value that is even smaller than c, and by the Archimedean Property, it must be 0. This is not a very rigorous proof though, since I didn't really use continuity.