nLab real polynomial function

Definition

A real polynomial function is a polynomial function in the real numbers, a function $f:\mathbb{R} \to \mathbb{R}$ with a natural number $n \in \mathbb{N}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ which satisfy one of these conditions:

• for all $x \in R$,

  $$f(x) = \sum_{i:[0, n)_\mathbb{N}} a(i) \cdot x^i$$

  where $x^i$ is the $i$-th power function for multiplication.

• $f$ is a solution to the $n$-th order linear homogeneous ordinary differential equation

  $$\frac{d^n f}{d x^n} = 0$$

  with initial conditions

  $$\frac{d^i f}{d x^i}(0) = i! \cdot a(i)$$

  for each natural number $i:[0, n)_\mathbb{N}$

Properties

Degree

The degree of a real polynomial function is defined as the maximum natural number $i \lt n$ such that $\vert a_i \vert \gt 0$. In constructive mathematics, there exist real polynomial functions for which one cannot prove that a particular natural number $i \lt n$ is the degree of the real polynomial function: i.e. the function sub-$\mathbb{R}$-algebra of real polynomial functions is not a Euclidean domain.

Pointwise continuity

Given a real polynomial function $f:\mathbb{R} \to \mathbb{R}$, $f$ is a pointwise continuous function with respect to its metric topology defined through the absolute value function, subtraction, and its strict linear order.

Pointwise differentiability

Given a real polynomial function $f:\mathbb{R} \to \mathbb{R}$, $f$ is a pointwise differentiable function with respect to its metric topology defined through the absolute value function, subtraction, and its strict linear order.

Smoothness

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 $n$-th derivative of a smooth function is a smooth function. The zero function $f(x) = 0$ is a smooth function, every real polynomial function is a pointwise differentiable function and the $n$-th derivative of a polynomial function, with a natural number $n \in \mathbb{R}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ such that one of the two conditions above is satisfied, is the zero function. Thus, every polynomial function is a smooth function.

See also

• real number

• polynomial function

• real quadratic function

• real cubic function

• rational functions are continuous

References

The proof of pointwise continuity using epsilontic analysis is spelled out for instance in

• Kyle Miller, Polynomials are continuous functions, 2014 (pdf)

See also:

• Wikipedia, Polynomial function