In CPFSK, suppose we transmit a continuous bit stream of 1, then for how long will the frequency go on increasing?
You seem to misunderstand the mathematical formula given here.

In a system with two symbols, there will only be two frequencies, one indicating a '1' has been sent, one indicating that a '0' has been sent. In that system, the integral increases the argument of the cosine by $$D_{f} \cdot t$$ when sending a '1', which gives a cosine with frequency $$f_c + D_f $$
compared to a cosine with frequency $$f_c$$ when sending a '0'.

In a more general system with N symbols, there will be N possible frequencies.

There will never be a phase discontinuity, which is why it's called "Continuous Phase FSK".