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/ $\displaystyle d(0, f(x)) \leq \epsilon$. We can find a $\displaystyle y_1 $ such that $\displaystyle d(0,f(y_1))\leq \epsilon /2$. Suppose there was minimal value c, such that $\displaystyle f(c) \geq 0.$ 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.
There is a theorem say that “A continuous function on a closed interval has a minimum and a maximum”.
So $\displaystyle \left( {\exists m \in I} \right)\left( {\forall z \in I} \right)\left[ {f(m) \leqslant f(z)} \right]$.
So suppose that $\displaystyle \left( {\forall z \in I} \right)\left[ {0 \ne f(z)} \right]$ then from the given $\displaystyle f(m) > 0$.
A contradiction follows at once from the minimum.
You are simply over-thinking all of this.
If the absolute minimum is $\displaystyle f(m)$ there cannot be any $\displaystyle y$ such that $\displaystyle \left| {f(y)} \right| \leqslant \frac{1}{2}\left| {f(m)} \right|$
Otherwise, $\displaystyle f(m)$ would not be the absolute minimum!