nLab difference quotient

Definition

Given a Heyting field $F$ and a subtype $S \subseteq F$, let us define the subtype $\Delta_{\#}(S) \subseteq S \times S$ of pairs of elements apart from the diagonal as

$$\Delta_{\#}(S) \coloneqq \sum_{(x,y):S \times S} x \# y$$

In a field

Given a function $f:S \to F$, the difference quotient $q:\Delta_{\#}(S) \to F$ is the partial binary function

$$q(x, y) := \frac{f(x) - f(y)}{x - y}$$

In a vector space

Given a $F$-vector space $V$ and a function $f:S \to V$, the difference quotient $q:\Delta_{\#}(S) \to V$ is the partial binary function

$$q(x, y) := \frac{1}{x - y} (f(x) - f(y))$$

See also

• limit of a binary function approaching a diagonal

• differentiable function

• Newton-Leibniz operator

• rational function

• reciprocal function