Functions with primitive functions are not necessarily continuous, right?

Uh, first of all, this question is a strange question: "A function with an original function is not necessarily continuous", the condition is that a function with an original function, the conclusion is that the function (the function with the original function, that is, the derivative function) is not necessarily continuous, not rigorous enough, and the concept is vague; Then the first time it is incorrect, the derivative function is continuous, the second sentence is "continuous in the defined domain" uh, inevitably, the last sentence is very wrong, how can the existence of a small interval be deduced to exist in a large interval There are many counterexamples in the textbook; The second time I asked, "As long as there is a function of the original function, it must be continuous within the defined domain", does this definition domain refer to the original function or the derivative function?

Seeing the last time I realized what you wanted to ask, it is equivalent to asking "The original function is continuous (in the defined domain), and its derivative is not necessarily continuous (in the defined domain of the original function)" And the derivative function is not necessarily continuous There are two cases, (1) it is not necessarily derivable everywhere, and the defined domain is a true subset of the original function, (2) it is derivable everywhere, but the derivative function has a discontinuity; It is easy to prove by the counter-proof method that "the original function is continuous, and its derivative must be continuous":(1)y=|x|continuous, but its derivative function has no defined domain at x=0; (2) The piecewise function y= (1-x 2)(-1 x 1), y=f(x) Other, the original function is continuous but its derivative is interrupted at x=1, -1. Any example of (1) and (2) can be used as a counterexample to the original proposition, so that "the original function is continuous (in the defined domain), and its derivative is not necessarily continuous (in the defined domain of the original function)".

First, the original function must be continuous (the property is arbitrary x derivable), and the derivative of the derivative function must be continuous within the derivable domain of the original function.

Conversely, the derivative must be continuous within the defined domain of the original function.

So this statement is not true.

NopeContinuous functionsNopeOriginal function。Because a continuous function must have an original function, a function is not a continuous original function.

Derived function. There can only be a second type of discontinuity, so if the function has a first type of discontinuity.

There must be no primitive function. Functional zhuan numbers with second-class discontinuities may or may not have the original function of the macro. For example, f(x)=x 2sin1 x, when x is not 0; f(0)＝0。

Easy to calculate f'(0)=0，f'Rent (x) = 2xsin1 x cos1 x, f at x 0'(x) There are discontinuities of the second type, f'(x) There are primitive functions. Another example is f(x)=1 x, when x is not equal; f(0)=0, this function has no original function.

Due toPiecewise functionsThe concept is too broad, and the textbook cannot clearly give the definition of the piecewise function in words, so it appears in the form of more practical examples.

Known function f(x) = Find the value of f(3).

Solution: From 3 (6), we know that f(3)=f(3+2)=f(5), and 5 (6), so f(5)=f(5+2)=f(7)

Again by 7 [6,+ so f(7)=7 2=5, therefore, f(3)=5.

The method of finding the function value of a piecewise function is to first determine the argument variable of the required value.

which segment it belongs to, and then press the expression of that segment.

Evaluate until the value is calculated.

The above content refers to: Encyclopedia - original function.

Continuous functions must have original functions, and discontinuous functions do not exist.

A derivative function can only have a second-class discontinuity, so if a function has a first-class discontinuity, there must be no original function. A functional zhuan number with a second-class discontinuity may or may not have the original function. For example, f(x)=x 2sin1 x, when Qin Shen x is not 0; f(0)＝0。

Easy to calculate f'(0)=0，f'(x) = 2xsin1 x cos1 x at x 0 f'(x) There are discontinuities of the second type, f'(x) There are primitive functions. Another example is f(x)=1 x, when x is not equal; f(0)=0, this function does not have the imitation number of the original family head.

Discontinuous functions may also have original functions.

As long as it's an integrable.

After integrating it.

You can get the original function of the slag.

So the functional formula is not continuous even if Bu Chun.

Don't judge from that.

Divide the situation. Assuming that the piecewise function hx=, which is very congratulatory and obvious, hx is not continuous at x=0, but belongs to the second type of discontinuity, and the source number of this function still has the original function fx.

The original collapse function is the piecewise function fx=

There must be one. Ant YuContinuous functionsThere must be a primitive function.

is an important theorem for the existence of the original function. A common way to prove this theorem is to construct a variable upper limit integral.

Proof is made using the definition of the derivative.

Therefore, if a function has a primitive function, there are many protople functions, and the concept of the primitive function was proposed to solve the inverse operation of derivation and differentiation. The problem of the existence of the original function is a fundamental theoretical problem in calculus, and when f(x) is a continuous function, its original function must exist.

Features of the original function:The function f(x) is a function defined in an interval, and if there is a derivative function f(x) such that any point in the interval has df(x)=f(x)dx, then the function f(x) is said to be the original function of the function f(x) in that interval. For example:

sinx is the original function of cosx.

Yes.

The original function must be continuous because the original function has a derivative.

So the original function must be continuous.

The original function refers to the function f(x) that is defined in a certain interval, if there is a derivative function f(x) such that df(x)=f(x)dx exists at any point in the interval, then the function f(x) is said to be the original function of the function f(x) in that interval.

Proto-function existence theorem:

If the function f(x) is continuous in a certain interval, then f(x) must exist in the interval, which is a sufficient but not necessary condition.

Also known as the "original function existence theorem".

Any function in the function family f(x)+c (c is any constant) must be the original function of f(x), so if the function f(x) has a primitive function, then its original function is infinitely multiple.

For example, x3 is a primitive function of 3x2, and it is easy to know that x3+1 and x3+2 are also primitive functions of 3x2. Therefore, if a function has a primitive function, there are many primitive functions, and the concept of primitive functions was proposed to solve the inverse operation of derivation and differentiation.

For example, if the velocity of an object moving in a straight line is known to be v=v(t) at any time, the law of its motion is required to find the original function of v=v(t). The existential problem of the original function is calculus.

The basic theoretical problem when f(x) is a continuous function.

, the fibrillary deficit function must exist.

NopeContinuous functionsNopeOriginal function。Because a continuous function must have an original function, a function is not a continuous original function.

If the function is integrable, then the function has an original function, and the original function is continuous, so for the function with only the first type of discontinuity, the original function exists and is continuous, and for the function with the second type of discontinuity, it needs to be analyzed on a case-by-case basis.

Related introductions. For continuity, there are many phenomena in nature, such as the change of temperature, the growth of plants, etc., which are continuously changing, and the reflection of this phenomenon in the relationship between functions is the continuity of functions.

at the limit of the function.

It has been emphasized that whether there is a limit to f(x) when x x0 has nothing to do with whether f(x) is defined at the point x0. But since the function is continuous at x0, it means that f(x0) must exist, and obviously δy=0 when δx=0 (i.e., x=x0<. Therefore, 0<|. can be canceled in the above derivation processδx|this condition.

1. Continuous functions must have original functions.

Second, when the function is discontinuous, it is known from Dab's theorem that if there is an original function in a discontinuous function, then the discontinuity point of this function is not a discontinuity point, the second is not a jump discontinuity point, and the third is not an infinite discontinuity point.

3. Discontinuous functions with ** discontinuity points do not necessarily exist as original functions, such as the piecewise tremor function.

f(x)=(1 x)*(sin1 x), (when x is not equal to 0); f(x)=0, (when x=0).The piecewise function f(x) has a break point x=0, but f(x) does not have the original function on any of the intervals [a,b] containing x=0 points.