Math Amateur
Gold Member
MHB
- 3,920
- 48
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 $$x$$ with $$b \gt x \gt \beta$$ and such that $$f(x) \lt 0$$ ... ... "My question is as follows:
How can we be sure that $$f(x) \lt 0$$ given $$x$$ with $$b \gt x \gt \beta$$ ... indeed how do we show rigorously that for $$x$$ such that $$b \gt x \gt \beta$$ we have $$f(x) \lt 0$$ ...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 $$x$$ with $$b \gt x \gt \beta$$ and such that $$f(x) \lt 0$$ ... ... "My question is as follows:
How can we be sure that $$f(x) \lt 0$$ given $$x$$ with $$b \gt x \gt \beta$$ ... indeed how do we show rigorously that for $$x$$ such that $$b \gt x \gt \beta$$ we have $$f(x) \lt 0$$ ...Help will be much appreciated ...
Peter