How about a function that is not continuous

Oh, it's actually very simple, the idea is the dirichlet function, that is.

When x is a rational number, f(x)=1, and when x is an irrational number, f(x)=0, it is clear that the function is discontinuous everywhere.

Then we make a little modification to it, we can satisfy only at one point continuously, and change the method to:

f(x)=x-a when x is a rational number, and f(x)=0 when x is irrational, where a is a rational number.

Then f(x) is only continuous at point a.

You just have to think about it.

Similarly, we can extend a function that is only at two points and continuous at only three points. By replacing f(x)=x-a at a rational point with f(x)=(x-a)(x-b)(x-c), we get a function that is continuous at only three points: a, b, and c.

Of course, continuous functions are not necessarily derivable.

f (x) = n =0 bncos (cn x) where a is a positive odd number , 01 +32 proves that the function is continuous everywhere but inderivable everywhere. Reference.

For example, a piecewise function is not a continuous function for a long time, as long as the function has no derivative, it is a discontinuous function, that is, a discontinuous function.

A piecewise function is a point where the limit value is not equal to the value of the function, and it is not continuous.

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=

Definition of function continuity: Let the function f(x) be defined in a certain neighborhood of the point x0, if lim(x x0)f(x)=f(x0), then f(x) is said to be continuous at the point x0.

If the function f(x) is continuous at every point in interval i, then f(x) is said to be continuous on interval i.

It is sufficient to determine that the derivative of the function is continuous, and if it is derivable, it must be continuous.

Extended Resources:

The function y=f(x) When the change in the independent variable x is small, the change in the dependent variable y is also small. For example, the temperature changes over time, and as long as the time changes very little, the change in temperature is also very small; Another example is that the displacement of a free-fall body changes with time, and as long as the time change is short enough, the change in displacement is also very small.

For this phenomenon, we say that the dependent variable is continuously changing with respect to the independent variable, and the image of the continuous function in a Cartesian coordinate system is a continuous curve with no breaks. From the nature of the limit, it can be seen that the sufficient and necessary condition for a function to be continuous at a certain point is that it is continuous at that point to the left and right.

As for continuity, there are many phenomena in nature, such as changes in temperature and plant growth. The reflection of this phenomenon in the relationship between functions is the continuity of functions.

Let the function <>

At the point <>

is defined in a certain neighborhood, and if there is <> then the function is said to be at the point <>

and is called <>

is a continuous point of the function.

Let the function be in interval <>

There are definitions, such as side effects<>

In <>

The left limit of is present and equal to <>i.e., <> then the function is said to be <> at the point

Left continuous. Let the function be in interval <>

There is a definition if <>

In <>

where the right limit exists and is equal to <>i.e., :, then the function <>

At the point <>

Right continuous.

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.

The sum, difference, product, and quotient of a finite number of continuous functions (with a denominator of non-zero) are continuous functions.

Proof: It is only necessary to use the limit algorithm to find f(x)*g(x) 0 or when x tends to x. , k(x) = f(x. ）*g（x。That's it.

The inverse of a continuous monotonically increasing (decreasing) function, which is also continuously monotonically increasing (decreasing); The composite function of a continuous function is continuous.

No matter what kind of function, as long as there is a primitive function, the primitive function must be derivable, and therefore must be continuous. Piecewise functions

, the original function obtained by the piecewise integral is also piecewise .

The original function refers to the function f(x) of the function f(x) that is defined in a certain interval of the occult penetration, 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.

If the function f(x) is continuous over an interval, then f(x) must have an original function in the bridge of repentance in that 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 an original function, then its original function is infinitely plentiful.

Continuous by all means! Recall the definition of the original function, the derivative of f(x) is equal to f(x), and f(x) is called a primitive function of f(x). It has been shown here that f(x) is derivable, and the unary function can be derived continuously, so the original function f(x) is a congratulatory chain that is fixedly continuous.

In fact, f(x) is not necessarily continuous. Bad albums.

The original function must be continuous.

The continuous destruction can be seen in a geometric sense, or it can be proved by other methods, but it is not continuous because it is derivable. For example, when a function has a finite number of hop breaks, the original function exists and is continuous, but not derivative.

So the original function, although continuous, is not necessarily derivable. (Of course, the premise is that you have to have the original function, which means you have to be integrable).

Judged by the definition of continuity of the function.

Function Continuity Definition:

For any x0 in the defined domain, there is limf(x)=f(x0)(x->x0) in the domain of x0

That is, when the limit value of the function at x0 is equal to the value of the function at that point, the function is continuous at that point, and if the function is continuous at every point in the defined domain, then the function is continuous in the defined domain.

The function is continuous, but not continuous"This sentence may be misexpressed or there is some misunderstanding. In mathematics, continuity and continuous functions are the same concept.

A function is called a continuous function, which means that within its defined domain, the value of the function changes with the change of the independent variables, and there are no jumps or breaks at each point. In other words, if the image of a function can be drawn by a pen without leaving the paper, then it is a continuous function.

Conversely, if a function has breaks or jumps at certain points, then it is not a continuous function.

Therefore, functional continuity and continuous function are the same concept, and one should not say "continuous function but not continuous function". It could be that there was a mistake or confusion in the expression.

The function is g(x)=x2sin1x,x≠0g(x)=x2sin 1x,x≠0

Define the function g(x)=x2sin1x,x≠0g(x)=x2sin 1x,x≠0 on [0,1][0,1].

Supplementary definition g(0)=0g(0)=0, then the function g(x)g(x) is a continuous function, and the figure is as follows.

The derivative can be obtained as g (x)=2xsin1x cos1x,x≠0g (x)=2xsin 1x cos 1x,x≠0

And g (0) = 0g (0) = 0, so g (x) g (x) is not continuous at x = 0x = 0. A derivative exists, but is not a continuous function on rr.

Let all rational numbers between the closed interval [0,1][0,1], then function.

f(x)=∑n=0∞12ng(x−rn)f(x)=∑n=0∞12ng(x−rn)

Converges consistently at [0,1][0,1].

f′(x)=∑n=0∞12ng′(x−rn)f′(x)=∑n=0∞12ng′(x−rn)

The rational points rnrn on [0,1][0,1] are discontinuous, and the irrational points on [0,1][0,1] are continuous.