# Intermediate Value Theorem .... Browder, Theorem 3.16 .... ....

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In summary, the conversation discusses a question regarding the proof of Theorem 3.16 in Andrew Browder's book "Mathematical Analysis: An Introduction". The question is about demonstrating the existence of a delta value such that for all t within a certain interval, the function g(t) is greater than a certain value, given that f(b) is greater than a certain value. The answer provided involves using the sign-preserving property of continuous functions and shows that the existence of such a delta value can be proven.
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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Theorem 3.16 ...Theorem 3.16 and its proof read as follows:
View attachment 9549
View attachment 9550
In the above proof by Andrew Browder we read the following:

" ... ... But $$\displaystyle f(b) \gt y$$ implies (since $$\displaystyle f$$ is continuous at $$\displaystyle b$$) that there exists $$\displaystyle \delta \gt 0$$ such that $$\displaystyle f(t) \gt y$$ for all $$\displaystyle t$$ with $$\displaystyle b - \delta \lt t \leq b$$. ... ... "My question is as follows:

How do we demonstrate explicitly and rigorously that since $$\displaystyle f$$ is continuous at $$\displaystyle b$$ and $$\displaystyle f(b) \gt y$$ therefore we have that there exists $$\displaystyle \delta \gt 0$$ such that $$\displaystyle f(t) \gt y$$ for all $$\displaystyle t$$ with $$\displaystyle b - \delta \lt t \leq b$$. ... ... Help will be much appreciated ...

Peter

***NOTE***

The relevant definition of one-sided continuity for the above is as follows:

$$\displaystyle f$$ is continuous from the left at $$\displaystyle b$$ implies that for every $$\displaystyle \epsilon \gt 0$$ there exists $$\displaystyle \delta \gt 0$$ such that for all $$\displaystyle x \in [a, b]$$ ...

we have that $$\displaystyle b - \delta \lt x \lt b \Longrightarrow \mid f(x) - f(b) \mid \lt \epsilon$$

#### Attachments

• Browder - 1 - Theorem 3.16 ... ... PART 1 ... .png
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• Browder - 2 - Theorem 3.16 ... ... PART 2 .png
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We have $f(b) > y$ or $f(b)-y >0$. Define $g(b) = f(b)-y>0$.

Since $f$ is continuous at $b$, $g$ is continuous at $b$.

Thus for all $\epsilon_0 >0$, there exist $\delta_0 >0$ s.t. $t \in [a,b]$ and $|t-b|<\delta_0$ imply $|g(t)-g(b)| < \epsilon_0$.

In particular, for $\epsilon := g(b)/2>0$ there exists $\delta >0$ s.t. $t \in [a,b]$ and $|t-b|<\delta$ imply $|g(t)-g(b)| < g(b)/2$.

Therefore $-g(b)/2 <g(t)-g(b) < g(b)/2$. Hence $g(t)> g(b)-g(b)/2 = g(b)/2$.

Rewriting $g(t) > g(b)/2$ in terms of $f$ we have $f(t) > \frac{1}{2}(f(b)+y)> \frac{1}{2}(y+y) = y$ as $f(b)>y$.

Thus $f(t)>y$ whenever $t \in [a,b]$ and $|t-b|<\delta$; that's $t \in [a,b] \cap (b-\delta, b+\delta)$, i.e. whenever $b-\delta < t \le b.$

This calculation can be skipped by appealing to sign-preserving property of limits/continuous functions:

Sign-preserving property (for continuous functions): Let $f: I \to \mathbb{R}$ be continuous at $c \in I \subseteq \mathbb{R}$.

1. If $f(c)>0$ then there exists $M>0$ and $\delta >0$ s.t. $x \in I$ and $|x-c|< \delta$ implies $f(x) > M$.

2. If $f(c)<0$ then there exists $N<0$ and $\delta >0$ s.t. $x \in I$ and $|x-c|< \delta$ implies $f(x) <N$.

Last edited:
MountEvariste said:
We have $f(b) > y$ or $f(b)-y >0$. Define $g(b) = f(b)-y>0$.

Since $f$ is continuous at $b$, $g$ is continuous at $b$.

Thus for all $\epsilon_0 >0$, there exist $\delta_0 >0$ s.t. $t \in [a,b]$ and $|t-b|<\delta_0$ imply $|g(t)-g(b)| < \epsilon_0$.

In particular, for $\epsilon := g(b)/2>0$ there exists $\delta >0$ s.t. $t \in [a,b]$ and $|t-b|<\delta$ imply $|g(t)-g(b)| < g(b)/2$.

Therefore $-g(b)/2 <g(t)-g(b) < g(b)/2$. Hence $g(t)> g(b)-g(b)/2 = g(b)/2$.

Rewriting $g(t) > g(b)/2$ in terms of $f$ we have $f(t) > \frac{1}{2}(f(b)+y)> \frac{1}{2}(y+y) = y$ as $f(b)>y$.

Thus $f(t)>y$ whenever $t \in [a,b]$ and $|t-b|<\delta$; that's $t \in [a,b] \cap (b-\delta, b+\delta)$, i.e. whenever $b-\delta < t \le b.$

This calculation can be skipped by appealing to sign-preserving property of limits/continuous functions:

Sign-preserving property (for continuous functions): Let $f: I \to \mathbb{R}$ be continuous at $c \in I \subseteq \mathbb{R}$.

1. If $f(c)>0$ then there exists $M>0$ and $\delta >0$ s.t. $x \in I$ and $|x-c|< \delta$ implies $f(x) > M$.

2. If $f(c)<0$ then there exists $N<0$ and $\delta >0$ s.t. $x \in I$ and $|x-c|< \delta$ implies $f(x) <N$.

Now working through what you have written ...

Thanks again ...

Peter

## 1. What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a mathematical theorem that states that if a continuous function has values of opposite signs at two points, then there must be at least one point between them where the function has a value of zero.

## 2. Who discovered the Intermediate Value Theorem?

The Intermediate Value Theorem was first proved by French mathematician Bernard Bolzano in the early 19th century. However, it was later independently proven by German mathematician Karl Weierstrass in the mid-19th century and is now often referred to as the Bolzano-Weierstrass Theorem.

## 3. What is the significance of the Intermediate Value Theorem?

The Intermediate Value Theorem is a fundamental theorem in calculus and real analysis. It is used to prove the existence of solutions to equations and to show that continuous functions have certain properties. It is also used in the proof of other important theorems, such as the Mean Value Theorem and the Fundamental Theorem of Calculus.

## 4. Can the Intermediate Value Theorem be applied to all functions?

No, the Intermediate Value Theorem can only be applied to continuous functions. A function is considered continuous if it has no sudden jumps or breaks and can be drawn without lifting the pen from the paper. If a function is not continuous, the Intermediate Value Theorem does not apply.

## 5. How is the Intermediate Value Theorem used in real-world applications?

The Intermediate Value Theorem has many real-world applications, especially in the fields of engineering and science. For example, it can be used to predict the existence of roots of equations, such as in the design of bridges and buildings. It is also used in optimization problems, where finding the maximum or minimum value of a function is important, such as in economics and physics.

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