- #1

Adesh

- 735

- 191

- Homework Statement
- We have a function ##f:[0,1] \mapsto \mathbb R## such that

$$

f(x)= \begin{cases}

x& \text{if x is rational} \\

0 & \text{if x is irrational} \\

\end{cases}

$$

- Relevant Equations
- Upper Darboux sum is

$$

U(f,P) = \sum_{i}^{n} M_i (x_i - x_{i-1})

$$

We're given a function which is defined as :

$$

f:[0,1] \mapsto \mathbb R\\

f(x)= \begin{cases}

x& \text{if x is rational} \\

0 & \text{if x is irrational} \\

\end{cases}

$$

Let ##M_i = sup \{f(x) : x \in [x_{i-1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 , \cdots , x_n\}## we define the upper Darboux sum as

$$

U(f,P) = \sum_{i}^{n} M_i (x_i - x_{i-1})

$$

Now, I need your help in showing that ##M_i = x_i## for any interval ##[x_{i-1}, x_i]##. Please don't give complete solution, I want to learn it, I'm a beginner in Real Analysis.

Thank you!

$$

f:[0,1] \mapsto \mathbb R\\

f(x)= \begin{cases}

x& \text{if x is rational} \\

0 & \text{if x is irrational} \\

\end{cases}

$$

Let ##M_i = sup \{f(x) : x \in [x_{i-1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 , \cdots , x_n\}## we define the upper Darboux sum as

$$

U(f,P) = \sum_{i}^{n} M_i (x_i - x_{i-1})

$$

Now, I need your help in showing that ##M_i = x_i## for any interval ##[x_{i-1}, x_i]##. Please don't give complete solution, I want to learn it, I'm a beginner in Real Analysis.

Thank you!