MindMap Gallery limit and continuity
This is a mind map about limits and continuity, including comparison of infinitesimals, operations of continuous functions and continuity of elementary functions, etc.
Edited at 2023-11-11 21:33:35This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
limit and continuity
1.6 Limit Existence Criterion Two Important Limits
Clamping Criterion
monotonic bounded criterion
definition
criteria
A monotonic bounded sequence must have a limit
If a sequence of numbers decreases monotonically and has a lower bound, then it has a limit.
If a sequence of monotonically increasing numbers has an upper bound, then it has a limit.
Two important limits
1.7 Comparison of infinitesimals
Properties of infinitesimal equivalence relations
reflexivity
symmetry
Transitivity
Common equivalent infinitesimal
Equivalent infinitesimal substitution theorem
1.8 Continuity and discontinuity points of functions
continuity of function
definition
If the limit value is equal to the function value, it is continuous
function break point
definition
Discontinuities of the first kind
Can remove discontinuities
jump break point
Type II discontinuities
infinite discontinuity
Oscillation break point
1.9 Operations of continuous functions and continuity of elementary functions
Four arithmetic operations for continuous functions
Continuity of composite functions
If the original function is continuously monotonic, then the inverse function of the function is also continuously monotonic.
The composite function of a continuous function is continuous
Continuity of elementary functions
Basic elementary functions are continuous within their domain
All elementary functions are continuous within their definition interval
1.10 Properties of continuous functions on closed intervals
Boundedness and the Maximum and Minimum Theorem
Maximum Minimum Theorem
boundedness theorem
Zero point theorem and intermediate value theorem
1.5 Limit algorithm
Theorem 1
Corollary 1
Corollary 2
Theorem 2
1.4 Infinitely Small and Infinitely Large
infinitesimal
definition
theorem
properties of infinitesimal operations
The algebraic sum of a finite number of infinitesimals is still infinitesimal
The product of a bounded function and an infinitesimal is still infinitesimal
The product of a constant and an infinitesimal is infinitesimal
The sum of infinite infinitesimals is not necessarily infinitesimal
The product of a finite number of infinitesimals is infinitesimal
gigantic
definition
The relationship between infinity and infinitesimal
1.3 Limits of functions
The limit of a function when the independent variable approaches infinity
definition
theorem
The independent variables tend to the limit of the finite-valued function
definition
theorem
Properties of function limits
uniqueness
local boundedness
Local number preservation
The relationship between function limits and sequence limits
1.2 Limits of Sequences
Convergent sequence properties
uniqueness
Boundedness
Number retention
Any subsequence of a convergent sequence converges to the same sequence
The concept of sequence
Definition of sequence
The limit of a sequence
1.1 Function
definition
Domain, range, corresponding rule
Function operations
Arithmetic
Inverse function
Inverse function existence theorem
properties of functions
Boundedness
Monotonicity
parity
cyclical
A periodic function does not necessarily have a minimum positive period
elementary functions
Non-elementary functions must be piecewise functions, but piecewise functions are not necessarily non-elementary functions.
Mapping and surjection
Limit arithmetic rules for composite functions
At least one of the left and right limits does not exist
There are both left and right limits