Is there such a thing as a continuous phase BPSK?
You're right that the transitions can be abrupt, but you can always choose to use pulse shaping, i.e., instead of using a rectangular pulse, use some smoother function $p(t)$. The corresponding baseband signal is then

$$s(t)=\sum_kA_kp(t-kT)\tag{1}$$

where $A_k$ are the symbols, and $T$ is the symbol length. A smooth function $p(t)$ will avoid abrupt transitions, and will therefore reduce spectral sidelobe levels. I think this paper uses that idea.