What is a black hole with no event horizon? Does that mean that once crossing, you are instantly subject to a great gravity? So there's a gravity gradient? How is that possible?

It's a naked singularity. Light can escape from the center ring of the singularity and you could possibly observe it. What exactly it is we don't know, our physics just break down that deep into the gravity well. It could just be a barely visible ring that saws your spaceship in half if you fly through or a wormhole or possibly anything really.

The event horizon is not a shield from gravity. It's the theoretical shell/radius inside which light cannot escape no matter what direction it's travelling. Photons sort of accumulate around the event horizon of a spinning black hole[0], because it takes them more orbits to escape the closer they get to the horizon, but without any matter/photos to observe, the event horizon is just a theoretical shell/radius limiting observation.

A black hole without an event horizon is a naked singularity[1], a concept in theoretical physics resulting from looking at the mathematical equations that describe black holes and observing that the event horizon's radius becomes undefined at certain angular velocities[2]. It's not clear if it can exist in reality.

AFAIK (not a physicist), gravitational force of a specific object (as a function of distance) cannot be discontinuous, can it? It's definitely not under the classical F=G(M1)(M2)/r^2 equation. Force can't be discontinuous unless mass changes discontinuously or space changes discontinuously. But I don't know if there are extreme conditions (and a naked singularity might qualify) where that equation breaks down, or is thought to break down?

[0] to support the movie Interstellar, several people worked on an accurate rendering, based on known physics, of a rotating black hole: https://astronomynow.com/2015/02/14/interstellar-technology-...

[1] https://en.wikipedia.org/wiki/Naked_singularity

[2] https://physics.stackexchange.com/questions/147034/rotating-...