What is the relationship between derivable and continuous, and what is the relationship between deri

The relationship between continuity and guideliness, come and learn.

There are four more classic sentences about the derivatives and continuities of functions:

1. Continuous functions are not necessarily derivable.

2. A derivable function is a continuous function.

3. The higher the order derivative function curve, the smoother the curve.

4. There are functions that are continuous everywhere but non-derivable everywhere.

The existence and "equality" of the left and right derivatives is the sufficient and necessary condition for the function to be derivable at that point, not the left limit = the right limit (both the left and right limits exist). Continuity is the value of the function, derivability is the rate of change of the function, and of course the derivability is a higher level.

Of course, your derivation is not correct, but the sufficient and necessary conditions for derivability are: the left and right derivatives exist and are equal, that is, the left and right limits exist and are equal.

The sufficient and necessary conditions for continuous continuity are: left limit = right limit = value of the function at that point, that is, both the left and right limits exist and are equal and equal to the value of the function.

Therefore, it must be continuous, but continuous is not necessarily continuous.

The derivative is defined as δf(x) The limit of δx, and δf(x) must be the same or higher order infinitesimal of δx, so that δf(x) must be 0 when derivable, i.e., f(x) is continuous.

f(x) is derivable<=> f(x) is derivable at x0, and both the left and right derivatives exist and are equal.

f(x)=|x|

1,x>0

f(x)-f(0)]/(x-0)

1,x<0

When x - >0, the limit does not exist, so f is not derivable at the point x=0!

It must be continuous, and continuous may not be continuous. That is, there are functions that are continuous everywhere at least within a region, but not derivatives everywhere.

The continuous and derivable relationship of a function.

This is based on the definition of derivative.

The derivative of f(x) = [f(x+dx)-f(x)] dx, e.g. f(x)=1 x x is not equal to 0

As you can see from the graph of this function, it's obviously discontinuous, because the graph of the function is at x=0 so that there is none.

But it's really directive!!

Continuous, must be guided.

Leadible is not necessarily continuous.

Just these few words.

The relationship between continuous and derivable is: derivable must be continuous, and continuous is not necessarily derivable.

Continuity is a necessary condition for derivability, but it is not a sufficient condition, from the derivable can be deduced continuous, from continuous can not be deduced from the derivable. It can be said that because it is leadable, it is continuous. It cannot be said: because it is continuous, it is derivable.

Sufficient and necessary conditions for the derivability of the function.

The function is continuous at that point and both the left and right derivatives exist and are equal. Theorem of the relationship between derivability and continuity of functions: If the function f(x) is derivable at x0, it must be continuous at the point x0.

The above theorem states: the function is derivative, and the function is continuous; Function continuity is not necessarily derivable; Discontinuous functions must not be derivative.

In calculus, a function of a real variable is a derivative function if it exists at every point in the defined domain. Intuitively speaking, the function image is relatively smooth at every point in its definition domain, and does not contain any sharp points or breakpoints.

1.Consistent continuity theorem

If the function f(x) is continuous over the closed interval [a,b], then f(x) is consistently continuous over the closed interval [a,b].

2.Integrable conditions

1) Accumulation of necessary conditions.

Theorem If the function f(x) is integrable on [a,b], then f(x) must be bounded on [a,b].

2) Sufficient conditions for accumulation.

Theorem 1 If the function f(x) is continuous on [a,b], then f(x) is integrable on [a,b].

Theorem 2 If the function f(x) is bounded on [a,b] and has only a finite number of discontinuities, then f(x) is integrable on [a,b].

Theorem 3 If the function f(x) is monotonic on [a,b], then f(x) is integrable on [a,b].