Difference quotient

Definition

Algebraic definition

The difference quotient of a function between two distinct points in its domain is defined as the quotient of the difference between the function values at the two points by the difference between the two points.

In symbols, if $f$ is a function defined on some subset of the reals and $x_1,x_2$ are distinct elements in the domain of $f$, then the difference quotient of $f$ between $x_1$ and $x_2$, denoted $\mathrm{\Delta }f(x_1,x_2)$, is defined as:

$\mathrm{\Delta }f(x_1,x_2):=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

Note that the definition is symmetric in $x_1$ and $x_2$, i.e., we have:

$\mathrm{\Delta }f(x_1,x_2)=\mathrm{\Delta }f(x_2,x_1)$

Geometric definition

The difference quotient of a function between two distinct points in its domain is defined as the slope of the chord joining the corresponding points in the graph of the function.

In symbols, if $f$ is a function defined on some subset of the reals and $x_1,x_2$ are distinct elements in the domain of $f$, then the difference quotient of $f$ between $x_1$ and $x_2$ is defined as the slope of the line segment joining the points $(x_1,f(x_1))$ and $(x_2,f(x_2))$, both of which are part of the graph of $f$.

Definition as a function

Consider a function $f$ with domain a subset $S$ of $\mathbb{R}$. The difference quotient, denoted $\mathrm{\Delta }f$, is a function defined on $S\times S\setminus \mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}(S)$ where $\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}(S)$ is the diagonal subset $\{(x,x):x\in S\}$. In other words, the difference quotient is defined on the set of ordered pairs $\{(x_1,x_2):x_1,x_2\in S,x_1\ne x_2\}$. It is defined as:

$\mathrm{\Delta }f(x_1,x_2)=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

The function is symmetric, i.e., $\mathrm{\Delta }f(x_1,x_2)=\mathrm{\Delta }f(x_2,x_1)$. Therefore, we can only think of it as a function on unordered pairs, i.e., we can view $\mathrm{\Delta }f$ as a function on the set $(\genfrac{}{}{0ex}{}{S}{2})$ of unordered pairs of distinct elements of $S$.

Related notions

• Derivative is defined as a limit of the difference quotient as one point approaches the other.

Properties of the difference quotient function

Joint continuity

For a continuous function $f$, the difference quotient function is a continuous function in the sense of joint continuity.

Explicitly, given an interval $I$, and a continuous function $f$ on $I$, the domain of $\mathrm{\Delta }f$ is a union of two triangular regions in $I\times I$, namely the regions above and below the diagonal. The function is symmetric, so the description on either side gives the description on the other side. The claim is that $\mathrm{\Delta }f$ is continuous at every point in both triangular regions, or equivalently, that $\mathrm{\Delta }f$ is continuous on both triangular regions.