What is continuity, what is differentiable/differentiable, what is analytic/singular point, what is Riemann integral, what is Lebesgue integral, what is Lagrange mean value theorem, what is Lagrange condition, integral mean value theorem, differential mean value theorem, Taylor formula, Rolle theorem, definition of sequence and function limit, Green formula, Gauss formula, relationship between differentiability and continuity (unary and multivariate), zero point existence theorem. How to find the zero point of the function, how to find the extreme point of the function (the point where the derivative is 0, the optimization problem), why the direction of gradient is the fastest direction of function decline, what is derivative, what is differential, what is integral, what is chain rule, gradient, directional derivative and gradient decline, and what is series?

Continuity is an attribute of a function. A continuous function refers to a function in which the change of input is small enough and the change of output is small enough. Example: The function L(t) of hair length changing with time is continuous.

Derivative describes the derivative of a function at a certain point. Necessary and sufficient conditions for derivability ① The function is continuous at this point (derivability must be continuous) ② The left derivative and the right derivative exist and are equal.

Derivability must be continuous, and continuity is not necessarily derivable. For example, the absolute value function f(x) =|x| is not differentiable at x=0 (the left and right derivatives are not equal).

The analysis of a function at a certain point means that the function can be derived anywhere at that point and its neighborhood.

No point defined. For example, the pole of f(x) = 1/x is 0.

If the function f(x) is continuous in the closed interval [a, b] and reachable in the open interval [a, b], and f(a) = f(b), then x0∈(a, b) exists, so

f'(x0) = 0