Since all the exponential families are absolutely continuous, if part is trivial. However, I could not prove the only if part. My idea is to prove by contradiction, i.e. given an event $A$ such that $P(X\in A)=0$ for $X\sim F$, where $F$ is some distribution from the exponential families. We wish to show that $P(Y\in A)=0$ if $Y$ follows any absolutely continuous distribution. My idea is to assume the contrary and then create a random variable $Z$ from exponential family such that $P(Z\in A)=0$. My problem actually is that, I can not find a way to utilize the given information to get a contradiction. Please help.