functions - Does $ccdot f(x) = f(cx) implies f$ is linear?

A linear functions $f$ is a function that has the following two properties:

1. $c\cdot f(x) = f(cx)$



2. $f(x+y) = f(x) + f(y)$

But does $c\cdot f(x) = f(cx) \implies f$ is linear? Can we assume property $2$ above, if we have a function $f$ satisfying property $1$?