Is the derivative of a differentiable function necessarily continuous

Is the derivative of a differentiable function necessarily continuous

Your question is too general. It neither defines the domain nor limits the scope of the function! But you should mean "must the derivative of a derivable function be continuous in the derivable domain of the original function?" the answer is yes. The answer on the first floor must be wrong, because x = 0 is not in the function domain. The same error on the second floor, oblique

The derivative function is continuous. Must the original function be continuous?

Because the continuous function must have the original function, the integral upper bound function is an original function of the derivative function, and the tangent integral upper bound function must be continuous, the derivative function must be continuous, and the original function must be continuous

The derivative of function f (x) is continuous at point X___ Conditions are differentiable___ condition

The differentiability of function f (x) at point x is a continuous "sufficient" condition and a differentiable "sufficient and necessary" condition

What are the conditions for the differentiability of functions?

Function is in the domain,
The function is continuous at this point, and the left and right derivatives exist and are equal
(this definition comes from the existence and equality of left and right limits)

Necessary and sufficient conditions for derivability of function? We know that if a function is differentiable, the necessary condition is that the function is continuous? What about the necessary and sufficient conditions? Whether the uniform continuity of a function can be proved is a necessary and sufficient condition for the differentiability of a function, just like the Cauchy convergence criterion in the judgment of whether a sequence of numbers has convergence If yes, it's best to provide proof. If it can't be fully proved, it's also OK to give the conditions The first floor and the fourth floor seem to be the definition

If a function is differentiable, it must be continuous. If a function is continuous, it is not necessarily differentiable. For example, y = LXL, the necessary and sufficient condition for the differentiability of a function at a point is a continuous function, and the left and right limits at that point exist and are equal. Of course, Tongji textbooks say that the necessary and sufficient condition for the differentiability of a function is that the left derivative and the right derivative are equal, which is

Let f (x) be continuous at x = 0 and f (x) / X limit exist when x approaches 0. It is proved that f (x) is continuous and differentiable at x = 0 Why does limf (x) / X exist, denominator -- > 0, so limf (x) = 0?

Because if limf (x) is not equal to 0, the limit of F (x) / X does not exist
Let limf (x) = C ≠ 0
Then when X - > 0, f (x) / X tends to + ∞ or - ∞
That is, the f (x) / X limit does not exist