Proof of Theorem (0.2): Questions to Ask

[solakis1]

for the theorem (0.2) attached i have the following questions to ask:

1) In the case where f(xo)<0 ,the author justifies the contradiction he comes to,by writing that :

"every xε [xo,xo+δ) belongs to X"

How can we prove that?

2) In the case where f(xo)>0 ,the author justifies the contradiction he comes to,by assming that:

(xo-δ) is an upper bound for X.

How can we prove that?

 

[Opalg]

Re: bolzano theorem

for the theorem (0.2) attached i have the following questions to ask:

1) In the case where f(xo)<0 ,the author justifies the contradiction he comes to,by writing that :

"every xε [xo,xo+δ) belongs to X"

How can we prove that?

2) In the case where f(xo)>0 ,the author justifies the contradiction he comes to,by assming that:

(xo-δ) is an upper bound for X.

How can we prove that?

Both these things come from the definition of $X$, which says that if $x\in[a,b]$ satisfies $f(x)\leqslant0$ then $x\in X$. (But note that in both cases the author seems to be assuming that $\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$)

 

[solakis1]

Re: bolzano theorem

Both these things come from the definition of $X$, which says that if $x\in[a,b]$ satisfies $f(x)\leqslant0$ then $x\in X$. (But note that in both cases the author seems to be assuming that $\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$)

yes,but how do you express mathematically the expression:

"$\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$"

 

[solakis1]

Re: bolzano theorem

Both these things come from the definition of $X$, which says that if $x\in[a,b]$ satisfies $f(x)\leqslant0$ then $x\in X$. (But note that in both cases the author seems to be assuming that $\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$)

O.K suppose he chooses delta small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$".

Then how does he get : SupX $\geq\delta+x_{o}> x_{o}$

 

[blue_raver22]

3) The author mentions using the Intermediate Value Theorem in the proof. Can you explain how this theorem is applied in the proof and why it is necessary?

4) In the proof, the author states that the function f is continuous on the interval [xo,xo+δ]. Can you explain why this is important in proving the theorem?

5) The theorem states that if f is continuous on an interval [a,b] and f(a) and f(b) have opposite signs, then there exists at least one root of f on the interval [a,b]. Can you provide an example to illustrate this theorem?

6) Is there a specific reason why the interval [a,b] is chosen for this theorem? Would the theorem still hold if a different interval was chosen?

7) The author mentions the concept of a root of a function. Can you explain what a root is and how it relates to the theorem being proved?

8) Are there any assumptions or conditions that need to be met for this theorem to hold true? If so, can you explain why they are necessary?

9) Can you provide any real-world applications or examples where this theorem would be useful in solving a problem or making a prediction?

10) Are there any other theorems or concepts related to this one that would be helpful to understand in order to fully grasp the proof and its implications?