I was going through Distribution function of the sum of poisson and uniform random variable. and my sir stated that below distribution function is continuous. \begin{align*} \mathbb P(X+Y \le a) = \sum_{i = 0}^{\lfloor a\rfloor-1} \mathbb P(X = i) + \mathbb P(X = \lfloor a\rfloor, Y \le a-\lfloor a\rfloor) \end{align*}
Here X is continuous and Y is discrete. How can we conclude that the distribution function is continuous?
By definition, a random variable (or its distribution) is continuous if its probability of any single value is always $0$.
If $X$ is discrete (with countable set of possible values $S$) and $Y$ is continuous, then for any $z$, $$ \mathbb P(X+Y=z) = \sum_{s \in S} \mathbb P(X=s,\; Y= z-s) = 0 $$ because $\mathbb P(X = s,\; Y = z - s) \le \mathbb P(Y = z-s) = 0$. Therefore $X+Y$ is continuous. Independence is not needed.