analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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A real polynomial function is a polynomial function in the real numbers, a function such that
is in the image of the function from the free monoid on , i.e. the set of lists of real numbers, to the function algebra , such that
is in the image of the canonical ring homomorphism from the real polynomial ring in one indeterminant to the function algebra , which takes constant polynomials in to constant functions in and the indeterminant in to the identity function in
There exists a natural number such that the -th order derivative of is equal to the zero function:
A real polynomial function is a function with a natural number and a list of length of real numbers which satisfy one of these conditions:
for all ,
where is the -th power function for multiplication.
is a solution to the -th order linear homogeneous ordinary differential equation
with initial conditions
for each natural number
Given a non-zero real polynomial function , the degree of is the maximum natural number where the -th derivative of is not the zero-function:
The degree of a real polynomial function, with additional data of a natural number and a list of length of real numbers , is defined as the maximum natural number such that .
In constructive mathematics, there exist real polynomial functions for which one cannot prove that a particular natural number is the degree of the real polynomial function: i.e. the function sub--algebra of real polynomial functions is not a Euclidean domain.
Given a real polynomial function , is a pointwise continuous function with respect to its metric topology defined through the absolute value function, subtraction, and its strict total order.
Given a real polynomial function , is a pointwise differentiable function with respect to its metric topology defined through the absolute value function, subtraction, and its strict total order.
Every real polynomial function is a smooth function. This could be shown coinductively: A smooth function is a pointwise differentiable function whose derivative is also smooth, and thus the -th derivative of a smooth function is a smooth function. The zero function is a smooth function, every real polynomial function is a pointwise differentiable function and the -th derivative of a polynomial function, with a natural number and a list of length of real numbers such that one of the two conditions above is satisfied, is the zero function. Thus, every polynomial function is a smooth function.
The proof of pointwise continuity using epsilontic analysis is spelled out for instance in
See also:
Last revised on August 21, 2024 at 01:52:29. See the history of this page for a list of all contributions to it.