Show that for $u_t(x,t)+u_x(x,t)=F(x,t)$, $u(x, x/c)=f(x)$ is a characteristic initial value problem if $f^\prime(x)=\frac{1}{c}F[x,\frac{x}{c}]$.

Question

Show that for $u_t(x,t)+u_x(x,t)=F(x,t)$, $u(x,\frac{x}{c})=f(x)$ is a characteristic initial value problem if $f^\prime(x)=\frac{1}{c}F(x,\frac{x}{c})$.

I did a problem similarly to this one, but I don't know if I can use the same method here. I'm kind'a lost. If someone could explain the criteria to the CIVP and explain how to do this, that would be great.