# Sup and inf of constant function

Problem: Show that the constant function is integrable and find its value of integration.

Suppose $f:\mathbb [a,b]\to \mathbb R$ such that $f(x)=\lambda$ where $\lambda$ is any constant. Let $P$ be any partition on $[a,b]$, ie $P=\left\{a=t_0 then Upper Darboux sum and Lower Darboux sum we evaluate by $U(f,P)=\sum_{1\leq k\leq n}\operatorname{Sup}\left\{f(x): x\in [t_{k-1},t_k]\right\}(t_k-t_{k-1})\\ L(f,P)=\sum_{1\leq k\leq n}\operatorname{inf}\left\{f(x): x\in [t_{k-1},t_k]\right\}(t_k-t_{k-1})$ Now what about the supremum and infimum of $f(x)$? If $\operatorname{sup}\left\{f(x): x\in[a,b]\right\}=\lambda$ but then $f(x)$ is constant so infimum of $f(x)$ is also $\lambda$ which immediately follows that $L(f,P)=\lambda(b-a)=U(f,P)$ Further $L(f)\geq L(f,P) ,\; U(f)\leq U(f,P) \implies L(f)=U(f)=\lambda(b-a)$ shows that $f(x)$ is integrable and its values is$L(f)\leq \int_a^b f(x) \leq U(f)\implies \int_a^b f(x) dx =\lambda(b-a)$

Now how to show that the supremum and infimum of the constant function is constant itself without using completeness property?

Any sorts of help will be appreciated.

Note by Naren Bhandari
1 week, 3 days ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

• bulleted
• list

1. numbered
2. list

1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$