I want to prove that if $X, Y$ are independent random variables and the distibution function of $X$ is continuous (it doesn't really matter wether this is for $Y$) then the distribution function of $X+Y$ is continuous. I have thought of using that $F$ continuous $\Leftrightarrow$ $F^{-1}$ is strictly increasing but cannot get to anything correct.
This is part of an exercise from my Probability Theory class.