If a function is continuous at a certain point, then it is bounded at this point, because if it is unbounded, then this point is its infinite discontinuous point, which contradicts continuity;

On the other hand, boundedness is not necessarily continuous, such as jump discontinuity;

If the function is continuous at a certain point, the left and right limits exist at that point and are equal to the function value at that point, so if it is continuous, the limit exists;

On the other hand, the existence of limit is not necessarily equal to the function value, that is, it is not necessarily continuous;

The function is bounded at a certain point, but the limit does not necessarily exist, such as oscillation discontinuity;

If a function has a limit at a certain point, it must be bounded, because if it is unbounded, the limit is infinite at most, and then the limit does not exist.

I hope I can help you!