Let $f(x):\mathbb{R}\to\mathbb{R}$ be a continuous function. For a scalar $a\in\mathbb{R}$, there exists a positive scalar $C$ such that the following condition is satisfied:
$$\|f(x)-af(y)\|\leq C\|x-ay\|$$
Is this a restrictive condition? Does this condition have a name? I know it is called Lipschitz condition when $a=1$. What are the properties of such functions. Any help is appreciated.
Suppose $a\ne 0$, and let $x=ay$. The inequality gives us that $f(ay)=af(y)$, for each $y\in\mathbb R$.
Substituting this back into the inequality, we have that $\lVert f(x)-f(ay)\rVert\le C\lVert x-ay\rVert$, which is just the Lipschitz condition.
The converse is also true, i.e., your condition holds iff $f$ is Lipschitz and for each $y\in\mathbb R$, $f(ay)=af(y)$.