Isotropic spherical mass stuck between two isotropic spherical greater masses

If you are working in Newtonian gravity (I assume you are, seeing the tag) and therefore you neglect all relativistic/quantum effects, then the central mass will stay in the same spot if placed exactly in the center of mass of the system.

The two bigger masses will be attracted towards the center by the small mass and the other big mass (and nothing stops them) while the small mass will be pulled in two opposite directions with the same magnitude, therefore (since the gravitational force, same as any force, is a vector) the two will cancel out.
For the conservation of angular momentum, if the system isn't rotating at the beginning it will continue to not rotate, and the movement will be completely linear along the axis passing through the center of the bodies.
As for the "is gravity continuous", it certainly is in Newtonian gravity, as well as in General Relativity (the best we can nowadays do to explain gravity). However, the question in a Quantum-Gravity environment is far from obvious: we just don't know.

Bonus: you can generalize this question with different, moving masses. It's the Lagrange points problem and the point you're asking about is L1 that can be found (numerically in general, analytically with some assumptions).