List of top Real Analysis Questions on Real Analysis

Let \( f: \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \to \mathbb{R} \) be given by \( f(x) = \frac{\pi}{2} + x - \tan^{-1}x \).
Consider the following statements:
P: \( |f(x) - f(y)| < |x - y| \text{ for all } x, y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \).
Q: \( f \) has a fixed point.
Then:

Let \( f_n: [0, 10] \to \mathbb{R} \) be given by \( f_n(x) = n x^3 e^{-n x} \) for \( n = 1, 2, 3, \dots \). Consider the following statements: P: \( (f_n) \) is equicontinuous on \( [0, 10] \).
Q: \( \sum_{n=1}^{\infty} f_n \) does NOT converge uniformly on \( [0, 10] \). Then:

For each \( x \in (0, 1) \), consider the decimal representation \( x = d_1 d_2 d_3 \cdots d_n \cdots \). Define \( f: [0, 1] \to \mathbb{R} \) by \( f(x) = 0 \) if \( x \) is rational, and \( f(x) = 18n \) if \( x \) is irrational, where \( n \) is the number of zeroes immediately after the decimal point up to the first nonzero digit in the decimal representation of \( x \). Then the Lebesgue integral \[ \int_0^1 f(x) \, dx = \(\underline{\hspace{1cm}}\). \]