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I am reading Cesar E. Silva's book entitled "Invitation to Real Analysis" ... and am focused on Chapter 4: Continuous Functions ...

I need help to clarify an aspect of the proof of Theorem 4.2.1, the Intermediate Value Theorem ... ...

Theorem 4.2.1 and its related Corollary read as follows:

View attachment 9562

View attachment 9563

In the above proof by Silva, we read the following:

" ... ... So there exists \(\displaystyle x\) with \(\displaystyle b \gt x \gt \beta\) and such that \(\displaystyle f(x) \lt 0\) ... ... "My question is as follows:

How can we be sure that \(\displaystyle f(x) \lt 0\) given \(\displaystyle x\) with \(\displaystyle b \gt x \gt \beta\) ... indeed how do we show rigorously that for \(\displaystyle x\) such that \(\displaystyle b \gt x \gt \beta\) we have \(\displaystyle f(x) \lt 0\) ...Help will be much appreciated ...

Peter

I need help to clarify an aspect of the proof of Theorem 4.2.1, the Intermediate Value Theorem ... ...

Theorem 4.2.1 and its related Corollary read as follows:

View attachment 9562

View attachment 9563

In the above proof by Silva, we read the following:

" ... ... So there exists \(\displaystyle x\) with \(\displaystyle b \gt x \gt \beta\) and such that \(\displaystyle f(x) \lt 0\) ... ... "My question is as follows:

How can we be sure that \(\displaystyle f(x) \lt 0\) given \(\displaystyle x\) with \(\displaystyle b \gt x \gt \beta\) ... indeed how do we show rigorously that for \(\displaystyle x\) such that \(\displaystyle b \gt x \gt \beta\) we have \(\displaystyle f(x) \lt 0\) ...Help will be much appreciated ...

Peter