So, let's say that I was given two independent random variables ξ and η and...

So, let's say that I was given two independent random variables $\xi$ and $\eta$ and was told that $\xi$ has continuous distribution (basically, $P(\xi =c)=0\mathrm{\forall }c\in \mathbb{R}$)
How can I prove that in such case their sum also will have continuous distribution? Problem that I have here is that I know that if two variables are independent then we can say $P_{\xi +\eta }=P_{\xi }\ast P_{\eta }$, where $\ast$ stands for measures convolution operation, but I am nor really sure how to compute such a convolution. Can you provide some explanation or intuition on how to calculate such integrals? In my case I need to know integral like this:
${\int }_{{\mathbb{R}}^2}{\mathbb{1}}_{\{c\}}(x+y)d{P}_{\xi }(x)d{P}_{\eta }(y)$
And I have no rigorous way of showing that it is actually zero.