If a function is differentiable at x0, it must be a continuous function at x0.
Definition of function derivability: (1) Let f(x) be defined at x0 and its vicinity, then when a tends to 0, if the limit of [f(x0+a)-f(x0)]/a exists, it is said that f(x) is derivable at x0.
(2) If any point (m, f(m)) on the interval (a, b) is derivable, it is said that f(x) is derivable on (a, b).
Conditions for differentiability of functions:
If the domain of a function is all real numbers, that is, the function is defined on it, can the function be differentiated everywhere in the domain? The answer is no, the differentiability of a function at a certain point in the domain requires certain conditions: the left and right derivatives of the function exist and are equal at this point. This is actually derived from a necessary and sufficient condition for the existence of limit (limit exists, and its left and right limits exist and are equal).
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