$ \frac{\ln(x)}{x} $ show this function is not continuous uniformly

Question

I tried with $x = \frac1n$ and $y = \frac1{n+1}$ but it gives me nothing. Same for $f(2x) - f(x)$.

Any idea?

Thank you

Answer 1

Hint:

Choose $x_n = 2/n$ and $y_n = 1/n$ where $|x_n - y_n| \to 0$ but

$$\left| \frac{\ln(2/n)}{2/n} - \frac{\ln(1/n)}{1/n}\right| \to \ldots$$

An unbounded derivative is not sufficient to prove non-uniform continuity as seen with the uniformly continuous function $f(x) = \sqrt{x}$ on $(0,\infty)$.