Is this Lipschitz continuous?

Question

Let $f$ be defined by $f(x)=x\log(1+\frac{a}{x})$, where $a$ is a positive scalar.

Is $f$ Lipschitz continuous?

Answer 1

The domain is $$D=(-\infty,-a)\cup (0,+\infty) $$

$f $ is differentiable at $D $ and $$f'(x)=\ln (1+a/x)-\frac {a}{x+a}$$

$f'$ is unbounded at $D $ thus it is not Lipschitz at $D $.

Answer 2

If $(-\infty, -a)$ is included in domain , then clearly $f$ is not Lipchitz since $$\lim_{x \rightarrow - a^-} f(x) = - \infty$$