In general, the continuity assumption of distribution functions at jumps is determined by convention. It could be right cont. or left cont. as long as it is used consistently. What you have is a non-zero prob. for the jump value. The main point is the precise definition of F(x) as a distribution function. You can have either of the following:
F(x)=P(X<x) - left continuous.
F(x)=P(X≤x) - right continuous.
Non-decreasing is required as well as the limit =1 at +∞.