Why is it continuous on the closed interval and derivable on the open interval

The derivability is derived from the limit, and the reason why it is the open-interval derivable is also made according to the expression of the derivable limit.

You can think of it this way, if it is derivable at the boundary of the closed interval, then how is the trend of change on the boundary reflected? What is outside the closed interval is not in the defined domain.

within. That is to say, the derivability on the boundary of the closed interval is indescribable, that is, it is meaningless. So in general mathematical analysis.

's textbook introduces "Neighborhood.

concept. Similarly, continuity over a closed interval is also in the service of limit derivation. The reason why it is a closed interval is that the continuity is more explicit, otherwise we cannot know the "starting point" and "end point" of the continuous function because we do not define the size of the "neighborhood" when we define the "neighborhood" (of course, it is impossible to give a definition of the specific size of the neighborhood).

If you can determine the "beginning" and "end" of a continuous function, you can get some definite properties, such as the mesovalue theorem.

Wait. At the same time, if the limit of the endpoint is outside the maximum and minimum range of the function, then the intermediate value theorem cannot be used, and other properties and solutions can be considered, such as constructing a new function, finding the endpoint that meets the requirements, etc.

If the function f(x) is derivable at every point in (a,b), then f(x) is said to be derivable on (a,b), then a derivative of f(x) can be established, referred to as the derivative, denoted f'(x)。

If f(x) is derivable in (a,b) and both the right derivative at the endpoint a of the interval and the left derivative at the endpoint b exist, then f(x) is said to be derivable on the closed interval [a,b], f'(x) is the derivative function on the interval [a,b], referred to as the derivative.

First, the function should be continuous over a closed interval, and there would be no point with a denominator of 0 in the derivative expression.

That is, there is a derivative at every point on the closed interval of the function.

On the closed interval, it can be deduced that the continuous on the closed interval.

g(x)=f(x)+x 3 is known from the properties of the elementary function that g(x) is continuously derivable in (0,1) on [0,1].

And g(0)=0,g(<0,g(1)=1 is caused by the existence of the continuous function mediator theorem (1) such that g( )=0

Applying Roll's theorem on [0, ] exists m (0, ) such that g'(m) = 0 i.e. f'(m)＋3㎡＝0

First inClosed intervalsOn the continuous is for useFermat's lemma

Secondly, in the general case of a point derivative, both the left and right derivatives exist and are equal.

So if it will beOpen sectionDerivable is replaced by a closed interval, then for the endpoint, the derivability becomes left and right, which is only a special case of derivability, and as a theorem, we need to describe the general situation, so use the open interval.

Roll's theorem, differential median theorem.

The generalized differential median theorem is that if an image of a function that is everywhere derivable and a horizontal line intersect at two different points.

Leaving aside the fact that f(x) is a constant function, a very important property used by the rolle theorem is the property that there is a maximum and minimum value in a closed-interval continuous function.

The Fermat theorem guarantees that f(x) is derivable at x0, and f(x0) is an extreme, then f'(x0) is equal to 0.

Here, changing the lead requires that it can be guided in the interval, but since the point you want is not taken at the end, you only need to open the interval.

As for why the closed interval continuous cannot be changed, it is because once it is changed to open interval continuous, the maximum and minimum value of Zaochun will be gone.

Because f(x) may be intermittent at x=a or x=b, the intermediate value may not be obtained.

The intermediate value theorem. Also known as the kizi median value theorem.

It is one of the properties of continuous functions on closed intervals, and one of the important properties of continuous functions in closed intervals. In mathematical analysis, the mesovalue theorem states that if a domain is defined.

is a continuous function f of [a,b], then at some point in the interval, it can take any value between f(a) and f(b).

Integrity

The theorem depends on, or is equivalent to, the completeness of real numbers. The mediator theorem does not apply to the first q of rational numbers, because there are irrational numbers between rational numbers.

For example, the function satisfies . However, there is no rational number x such that because is an irrational number.

The geometric meaning is overwhelming

Continuous curves on [a,b] with. In particular, if A is different from B, the continuous curve intersects the x-axis at least once. The "intermediate value theorem" is one of the properties of continuous functions over closed intervals.

Because the function is continuous on the closed interval, the left endpoint is required to be right continuous, and the right endpoint is left continuous; The derivability of a function requires that the left and right derivatives of the function exist and are equal at a point, and if it is a closed interval, it can only be verified whether there is a right derivative at the left endpoint and a left derivative at the right endpoint, so the function is not derivable at the endpoint of the closed interval.

Median value theorem. It is a point of a function or a slope of a function instead of the original function.

, so it is necessary to close the interval and continuously open the interval to be derivable.

Because the function is continuous on the closed interval, the left endpoint is required to be right continuous, and the right endpoint is left continuous; The derivability of a function requires that the left and right derivatives of the function exist and are equal at a point, and if it is a closed interval, it can only be verified whether there is a right derivative at the left endpoint and a left derivative at the right endpoint, so the function is not derivable at the endpoint of the closed interval.

Median value theorem. It is a point of a function or a slope of a function instead of the original function.

, so it is necessary to close the interval and continuously open the interval to be derivable.

A function is continuous (derivable) at any point in an interval when the function is continuous (derivable) at any point in the interval.

As for judging whether a function is continuous or derivable at a certain point, that is, whether a certain limit exists.

Determine whether the function f is continuous at the point x0, that is, determine whether the limit lim(x--x0)f(x) exists and is equal to f(x0).

Determine whether the function f is derivable at the point x0, that is, determine whether the limit lim(dx--0)(f(x+dx)-f(x)) dx exists.

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 <>

is continuous, and is called <>

is a continuous point of the function.

A function is in the open interval <>

is continuous at <>

continuous, if again in <>

Tap right consecutively, <

If the dots are left continuously, they are in the closed interval <>

Continuous, if continuous throughout the defined domain, is called a continuous function.

Obviously, from the nature of the limit, it is clear that the sufficient and necessary condition for a function to be continuous at a certain point is that it is continuous at that point.

Because the function is continuous on the closed interval, the left endpoint is required to be right continuous, and the right endpoint is left continuous; The derivability of a function requires that the left and right derivatives of the function exist and are equal at a point, and if it is a closed interval, it can only be verified whether there is a right derivative at the left endpoint and a left derivative at the right endpoint, so the function is not derivable at the endpoint of the closed interval.

Median value theorem. It is a point of a function or a slope of a function instead of the original function.

, so it is necessary to close the interval and continuously open the interval to be derivable.