How can I use this formula of the characteristic function in a discrete set: $$\phi_X(t)=\sum \exp(itx_k)P(X=x_k)$$
To prove that the characteristic function is defined, uniformly continuous and that the transformation $Y=aX+b$ gives $\phi_Y(t)=\exp(ibt)\cdot\phi_X(at)$ without the use of $E[e^{itx}]$ in the proof?
I can only find proofs online using $E[eitx]$ or proofs in a continuous set.
Denoting by $y_k$ the value taken by $Y$ when $X$ takes the value $x_k$ (therefore $P(Y=y_k)=P(X=x_k)$):
$$\Phi_Y(t)=\sum e^{ity_k}P(Y=y_k)=\sum e^{it(ax_k+b)}P(X=x_k)$$
$$\Phi_Y(t)=\sum e^{i(at)x_k}e^{itb}P(X=x_k)$$
$$\Phi_Y(t)=e^{itb}\underbrace{\sum e^{itax_k}P(X=x_k)}_{\Phi_X(at)}$$