nLab algebraic limit field

Idea

A field with a notion of a limit of a function that satisfy the algebraic limit theorems.

Definition

Let $F$ be a Heyting field and a Hausdorff function limit space, where $x^{-1}$ is another notation for the reciprocal function $\frac{1}{x}$. $F$ is a algebraic limit field if the algebraic limit theorems are satisfied, i.e. if the limit preserves the field operations:

• for all elements $c \in S$,

  $$\lim_{x \to c} 0(x) = 0$$

• for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that

  $$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
  $$\lim_{x \to c} f(x) + g(x) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$

• for all elements $c \in S$ and functions $f:S \to C$ such that

  $$\lim_{x \to c} f(x) = c$$
  $$\lim_{x \to c} -f(x) = -\lim_{x \to c} f(x)$$

• for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that

  $$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
  $$\lim_{x \to c} f(x) - g(x) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$$

• for all elements $c \in S$, integers $a \in \mathbb{Z}$, and functions $f:S \to C$, such that

  $$\lim_{x \to c} f(x) = c$$
  $$\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)$$

• for all elements $c \in S$ and $a \in S$, and functions $f:S \to C$, such that

  $$\lim_{x \to c} f(x) = c$$
  $$\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)$$

• for all elements $c \in S$,

  $$\lim_{x \to c} 1(x) = 1$$

• for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that

  $$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
  $$\lim_{x \to c} f(x) \cdot g(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$$

• for all elements $c \in S$, natural numbers $n \in \mathbb{N}$, and functions $f:S \to C$, such that

  $$\lim_{x \to c} f(x) = c$$
  $$\lim_{x \to c} {f(x)}^n = {\left(\lim_{x \to c} f(x)\right)}^n$$

• for all elements $c \in S$, and functions $f:S \to C$, such that

  $$\lim_{x \to c} f(x) = c$$

  if

  $$\lim_{x \to c} f(x) \# 0$$

  then

  $$\lim_{x \to c} {f(x)}^{-1} = {\left(\lim_{x \to c} f(x)\right)}^{-1}$$

• for all elements $c \in S$, and functions $f:S \to C$, such that

  $$\lim_{x \to c} f(x) = c$$

  if

  $$\lim_{x \to c} f(x) \# 0$$

  then

  $$\lim_{x \to c} f(x) \cdot {f(x)}^{-1} = 1$$

See also

• function limit space
• difference quotient
• algebraic limit vector space
• Newton-Leibniz operator
• differentiable space