Show that $x \cos(cx)$ is aperiodic

Question

I'm using the function $f(x)=x~cos(cx)$ in a paper and the periodicity of the function is relevant. Is there a simple way to show that a function of this form is aperiodic, or is it reasonable to simply state that it is aperiodic because of the multiplication with $x$?

By aperiodic I mean that in the following equation $T$ must be $0$:

$$x~\cos(cx) = (x+T) ~\cos(c(x+T)).$$

Thanks.

Answer 1

For $c=0$ it's obvious. If $c>0$ $$\lim_{n\to\infty }\frac{2\pi n}{c}\cos\left(c\frac{2\pi n}{c}\right)=+\infty $$ and $$\lim_{n\to\infty }\frac{\pi+2n\pi}{c}\cos\left(c\frac{\pi+2\pi n}{c}\right)=-\infty $$ therefore it's not periodic. For $c<0$ you the proof goes the same.