 #1
 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
$$
U(f,P) = \sum_{i}^{n} M_i (x_i  x_{i1})
$$
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_{i1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 , \cdots , x_n\}## we define the upper Darboux sum as
$$
U(f,P) = \sum_{i}^{n} M_i (x_i  x_{i1})
$$
Now, I need your help in showing that ##M_i = x_i## for any interval ##[x_{i1}, 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_{i1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 , \cdots , x_n\}## we define the upper Darboux sum as
$$
U(f,P) = \sum_{i}^{n} M_i (x_i  x_{i1})
$$
Now, I need your help in showing that ##M_i = x_i## for any interval ##[x_{i1}, x_i]##. Please don't give complete solution, I want to learn it, I'm a beginner in Real Analysis.
Thank you!