A function $\ell$ is slowly varying (in Karamata's sense) if it is a positive measurable function, defined on some neighbourhood $[X,\infty)$ of infinity, and satisfying $$ \ell(\lambda x)/\ell(x)\to1\quad\text{as}\quad x\to\infty $$ for each $\lambda>0$.
Is the difference $|\ell(\lambda x)-\ell(x)|$ bounded for large values of $x$, i.e. is there an $x_0$ such that the inequality $$ |\ell(\lambda x)-\ell(x)|\le c(\lambda) $$ is true for $x>x_0$?
We have that $|\ell(\lambda x)-\ell(x)|=\ell(x)|\ell(\lambda x)/\ell(x)-1|$. If $\ell$ is bounded, we are done. Hence, we need to investigate the case when $\ell(x)\to\infty$ as $x\to\infty$. We have an indeterminate form $\infty\times0$ in this case and we need to do something else. I can show that the inequality is true for some particular slowly varying functions (for example, the logarithmic function), but I have no idea how to proceed in the general case.
Any help is much appreciated!