Trouble understanding a step in the proof of the Intermediate Value Theorem

Question

I'm having some trouble understanding a particular proof of the intermediate value theorem. The theorem is stated as follows:

Assume that $f : [a,b]\rightarrow\mathbb{R}$ is continuous and that $f(a)$ and $f(b)$ have opposite signs. Then there is a point $c\in(a,b)$ such that $f(c)=0$.

The beginning of the proof is as follows:

Consider the case where $f(a)<00$, then $c

I don't see why $c0$, but I would really like to understand this, especially since the author seems to think it needs no further explanation. I did try looking at other proofs for the IMT, on this site and other sites, but none of them phrased the proof in this way and they didn't really further my understanding.

Any help would be greatly appreciated.

Answer 1

If $f$ is continous then there exists a ball with the center in $b$ such that $f(x) > 0$ in this ball. We could take for example the ball of the radius $\varepsilon$ such that $$\ |x-b|<\varepsilon \Rightarrow |f(x)-f(b)|

Such $\varepsilon$ exists according to continuity of $f$. So, then $c \le b - \varepsilon < b.$