# Continuity of f^+ .... Browder Corollary 3.13

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In summary, the proof for the continuity of f^+ at a point where f crosses the x-axis involves considering 3 cases and showing that in each case, there exists an open set that maps to an open set under f^+, thus proving the continuity of f^+. This proof is applicable for both points where f crosses the x-axis from above and below, and involves constructing a union of open sets to show that the preimage of an open set under f^+ is also open.
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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 with the proof of Corollary 3.13 ...Corollary 3.13 reads as follows:View attachment 9520Can someone help me to prove that if $$\displaystyle f$$ is continuous then $$\displaystyle f^+ = \text{max} (f, 0)$$ is continuous ...My thoughts are as follows:If $$\displaystyle c$$ belongs to an interval where $$\displaystyle f$$ is positive then $$\displaystyle f^+$$ is continuous since $$\displaystyle f$$ is continuous ... further, if $$\displaystyle c$$ belongs to an interval where $$\displaystyle f$$ is negative then $$\displaystyle f^+$$ is continuous since $$\displaystyle g(x) = 0$$ is continuous ... but how do we construct a proof for those points where $$\displaystyle f(x)$$ crosses the $$\displaystyle x$$-axis ... ..

Help will be much appreciated ...

Peter

#### Attachments

• Browder - Corollary 3.13 ... .png
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To prove $$f^+$$ is continuous at $$x_0$$ consider 3 cases:
1) $$f(x_0)> 0$$.
2) $$f(x_0)= 0$$.
3) $$f(x_0)< 0$$.

Since f is continuous, in case (1) there exist an interval around $$x_0$$ such that f(x)> 0
and $$f^+(x)= f(x)$$
for all x in the interval

Since f is continuous, in case (1) there exist an interval around $$x_0$$ such that f(x)< 0 and $$f^+(x)= 0$$ for all x in the interval.

HallsofIvy said:
To prove $$f^+$$ is continuous at $$x_0$$ consider 3 cases:
1) $$f(x_0)> 0$$.
2) $$f(x_0)= 0$$.
3) $$f(x_0)< 0$$.

Since f is continuous, in case (1) there exist an interval around $$x_0$$ such that f(x)> 0
and $$f^+(x)= f(x)$$
for all x in the interval

Since f is continuous, in case (1) there exist an interval around $$x_0$$ such that f(x)< 0 and $$f^+(x)= 0$$ for all x in the interval.
Thanks for the help ...

BUT ... you do not describe what to do in case 2 ... and as I indicated it is when f(x) = 0, specifically when f crosses the x-axis that I am having trouble dealing with ...

Peter

Peter said:
Thanks for the help ...

BUT ... you do not describe what to do in case 2 ... and as I indicated it is when f(x) = 0, specifically when f crosses the x-axis that I am having trouble dealing with ...

Peter

After reflecting on this problem for some time ... here is my proof for the situation where the point investigated is a point where $$\displaystyle f$$ crosses the x-axis ...I think it will suffice to prove that $$\displaystyle f^+$$ is continuous for the case where a point $$\displaystyle c_1 \in \mathbb{R}$$ is such that for $$\displaystyle x \lt c_1, \ f(x) = f^+(x)$$ is positive and for $$\displaystyle x \gt c_1, \ f^+(x) = 0$$ ... ... while for some point $$\displaystyle c_2 \gt c_1$$ we have that $$\displaystyle f^+(x) = 0$$ for $$\displaystyle x \lt c_2$$ and $$\displaystyle f(x) = f^+(x)$$ is positive for $$\displaystyle x \gt c_2$$ ... ... ... see Figure 1 below ...

View attachment 9522Now consider an (open) neighbourhood $$\displaystyle V$$ of $$\displaystyle f^+(c_1)$$ where...
$$\displaystyle V= \{ f^+(x) \ : \ -f^+(a_1) \lt f^+(c_1) \lt f^+(a_1)$$ for some $$\displaystyle a_1 \in \mathbb{R} \}$$

so ...$$\displaystyle V= \{ f^+(x) \ : \ -f^+(a_1) \lt 0 \lt f^+(a_1)$$ for some $$\displaystyle a_1 \in \mathbb{R} \}$$ ...

Then ... (see Figure 1) ...
$$\displaystyle (f^+)^{ -1 } (V) = \{ a_1 \lt x \lt a_2 \}$$ which is an open set as required ...
Further crossings of the x-axis by $$\displaystyle f$$ just lead to further sets of the nature $$\displaystyle \{ a_{ n-1 } \lt x \lt a_n \}$$ which are also open ... so ...

$$\displaystyle (f^+)^{ -1 } (V) = \{ a_1 \lt x \lt a_2 \} \cup \{ a_3 \lt x \lt a_4 \} \cup \ldots \cup \{ a_{n-1} \lt x \lt a_n \}$$
which being a union of open sets is also an open set ...

The proof is similar if $$\displaystyle f$$ first crosses the x-axis from below ...

Is that correct?Peter

#### Attachments

• Figure 1 - Continuity of f+ ... .png
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HallsofIvy said:
To prove $$f^+$$ is continuous at $$x_0$$ consider 3 cases:
1) $$f(x_0)> 0$$.
2) $$f(x_0)= 0$$.
3) $$f(x_0)< 0$$.

Since f is continuous, in case (1) there exist an interval around $$x_0$$ such that f(x)> 0
and $$f^+(x)= f(x)$$
for all x in the interval

Since f is continuous, in case (1) there exist an interval around $$x_0$$ such that f(x)< 0 and $$f^+(x)= 0$$ for all x in the interval.
TYPO: in the last sentence "case (1)" should have been "case (2)".

## 1. What is the Browder Corollary 3.13?

The Browder Corollary 3.13 is a mathematical theorem that states that if a function f^+ is continuous on a closed interval [a,b], then its upper Darboux integral is also continuous on [a,b]. This corollary is an extension of the Darboux's theorem on the continuity of integrals.

## 2. What is the significance of the Browder Corollary 3.13?

The Browder Corollary 3.13 is significant because it provides a necessary and sufficient condition for the continuity of upper Darboux integrals. This theorem is useful in many areas of mathematics, particularly in the study of real analysis and measure theory.

## 3. How is the Browder Corollary 3.13 related to continuity?

The Browder Corollary 3.13 is directly related to continuity as it states that if a function is continuous on a closed interval, then its upper Darboux integral will also be continuous on that interval. This means that the function's behavior remains consistent and predictable over the entire interval.

## 4. What is the difference between upper and lower Darboux integrals?

The upper and lower Darboux integrals are two ways of approximating the area under a curve. The upper Darboux integral uses the maximum value of the function on each subinterval, while the lower Darboux integral uses the minimum value. The Browder Corollary 3.13 specifically deals with the continuity of the upper Darboux integral.

## 5. How is the Browder Corollary 3.13 used in real-world applications?

The Browder Corollary 3.13 is used in real-world applications to analyze and model continuous functions. It is particularly useful in economics and physics, where continuous functions are used to represent real-world phenomena. This corollary helps ensure the accuracy and reliability of these models by providing a condition for the continuity of integrals.

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