## Problem 4.

Determine the continuous functions $f:(0, \infty) \rightarrow \mathbb{R}$ and the real constant $L$ knowing that:

a) $\lim _{x \rightarrow 0} x \cdot f(x)=L$ exists;
b) $\int_{x}^{2 x} f(t) d t=1,(\forall) x>0$.
## Solution and grading:

From b) we get $F(2 x)-F(x)=1,(\forall) x>0$, where $F$ is a primitive of $f$........2p

![](https://cdn.mathpix.com/cropped/2024_06_07_76895e800db888dc4a4dg-3.jpg?height=86&width=1442&top_left_y=1670&top_left_x=381)

![](https://cdn.mathpix.com/cropped/2024_06_07_76895e800db888dc4a4dg-3.jpg?height=82&width=1427&top_left_y=1758&top_left_x=383)
$L=\lim _{x \rightarrow 0} x \cdot f(x)=\lim _{n \rightarrow \infty} \frac{x}{2^{n}} \cdot f\left(\frac{x}{2^{n}}\right)=\lim _{n \rightarrow \infty} x \cdot f(x)=x f(x),(\forall) x>0$, from which $f(x)=\frac{L}{x},(\forall) x>0$.

By substitution into b) we get $L(\ln 2 x-\ln x)=1,(\forall) x>0$, from which $L=\frac{1}{\ln 2}$ and the function $f(x)=\frac{1}{x \ln 2},(\forall) x>0$.