How continuous phase variation is achieved in MSK scheme?
When the difference in the two carrier frequencies generated by a logical 1 and a logical 0 is half the data rate, the modulation index is 0.5 by definition of modulation index.

When the two carrier frequencies have this relationship then there are no sudden changes in phase as can be seen with plain ordinary FSK. Here's an example of MSK: -



The two carrier frequencies produced are 1200 Hz and 1800 Hz and have a difference of 600 Hz - this difference is half the data rate (1200 baud).

It's nothing more complicated than that.