Let's talk about limit and continuity first.

Limit is the most basic and the concept of continuity is based on it. Continuity is defined as follows. Let a function be defined in a neighborhood of a point. If the limit of function f(x) exists when x approaches x0, it is equal to its function value f(x0) at point x0.

Then we assume that the function f(x) is continuous at point x0. The concept of continuity is very tasteless. It's not in the syllabus and it's not explicitly tested. Just know, don't go too far.

That is to say, if a function has a limit at one point, it can be continuous, while the derivative is a tool and a function of slope, and its application range is very narrow. For example, if you are a linear function, you don't take the derivative at the inflection point, but you have limit and continuity. So the derivative only helps you find the limit, but it can't determine whether the limit exists. If there is a derivative, there must be a limit, there must be continuity, and if there is no derivative, there may be a limit and continuity. There is continuity, there must be a limit, there is a limit, and it is not necessarily continuous.