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How to judge the discontinuity?
First of all, we should know that the first type of discontinuity (both left and right limits exist) has the following two types: 1. The limits of functions on both sides of the jump discontinuity discontinuity are not equal. 2. The limits of functions on both sides of the de discontinuity discontinuity exist and the equal function is meaningless at this point. The second type of discontinuity (non first type of discontinuity) also has two kinds of 1 oscillation discontinuities
How to judge whether a second type of discontinuity is an infinite discontinuity or an oscillatory discontinuity
At least one of the left and right limits is infinite, and the discontinuity is infinite discontinuity
At least one nonexistent discontinuity in the left and right limits is the oscillation discontinuity
First type of discontinuity of derivative
If x0 is the discontinuous point of function f (x), but both the left limit and the right limit exist, x0 is called the first kind of discontinuity of function f (x). Relevant knowledge: let function y = f (x) be defined in a neighborhood of point x0, if the limit of function f (x) when x → x0 exists and is equal to its function value f (x0) at point x0, that is, LIM (x → x0
What do the first type of breakpoints and the second type of breakpoints mean
Removable breakpoint: the left limit and right limit of the function exist and are equal at this point, but not equal to the function value at this point or the function has no definition at this point. For example, the function y = (x ^ 2-1) / (x-1) is at point x = 1
Jump breakpoint: the left limit and right limit of the function exist at this point, but they are not equal. For example, the function y = |x| / X is at point x = 0. (Figure 2)
Infinite discontinuity: the function can be undefined at this point, and at least one of the left limit and right limit is ∞. For example, the function y = TaNx is at point x = π / 2
Removable discontinuities and jumping discontinuities are called type I discontinuities, also known as finite discontinuities. Other discontinuities are called type II discontinuities
From the above description of various discontinuities, it can be seen that function f (x) exists in both the left and right limits of the first type of discontinuities, while function f (x) does not exist in at least one of the left and right limits of the second type of discontinuities, which is also the essential difference between the first type of discontinuities and the second type of discontinuities
The second type of discontinuity in high numbers Let function f (x) be defined in a de centered neighborhood of point x0. On this premise, if function f (x) has one of the following three cases (1) Not defined at x = X0 (2) Although it is defined in x = x0, LIM (x → x0) f (x) does not exist (3) Although it is defined in x = x0 and lim (x → x0) f (x) exists, LIM (x → x0) f (x) ≠ f (x0) Then function f (x) is discontinuous at point x0, and point x0 is called discontinuous point or breakpoint of function f (x) What I want to ask is what kind of situation is that "(2) although it is defined in x = x0, LIM (x → x0) f (x) does not exist"? It seems that x = x0 has a definition, so f (x0) exists? For example
Let me help you sort out the three types of discontinuities corresponding to 1, 2 and 3 above: the second type of discontinuity (you understand that it is not called the second type, it is called jumping), the jumping discontinuity, and you can go to the discontinuity,
Among them, you ask, if x = x0 has a definition, then f (x0) exists?
Let me give an example, f (x) = 1 in [0,1)
F (x) = 2 in [1,2],
When x = 1, there is a definition but no limit. You can see that there is a jump
Why does the function have the second type of discontinuity in Higher Mathematics? There may be the original function
Examples are as follows:
Let f (x) = xsin (1 / x), X ≠ 0
0,x=0
Then f (x) = f '(x) = sin (1 / x) - (1 / x) cos (1 / x), X ≠ 0
When x = 0, f '(x) does not exist
It is easy to know that x = 0 is the second type of discontinuity of F (x), and f (x) has the original function f (x)
RELATED INFORMATIONS
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