# 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 $\Delta f(x_1,x_2)$, is defined as:

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

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

$\! \Delta f(x_1, x_2) = \Delta f(x_2,x_1)$

### 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 $\R$. The difference quotient, denoted $\Delta f$, is a function defined on $S \times S \setminus \operatorname{Diag}(S)$ where $\operatorname{Diag}(S)$ 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:

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

The function is symmetric, i.e., $\Delta f(x_1,x_2) = \Delta f(x_2,x_1)$. Therefore, we can only think of it as a function on unordered pairs, i.e., we can view $\Delta f$ as a function on the set $\binom{S}{2}$ 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.
• Divided differences are the generalization to more variables.

## Properties of the difference quotient function

### Symmetry

The difference quotient function is symmetric: for a function $f$ on a subset $S$ of $\R$, and for distinct points $x_1, x_2$ of $S$, we have:

$\Delta f (x_1,x_2) = \Delta f(x_2,x_1)$

### 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 $\Delta f$ 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 $\Delta f$ is continuous at every point in both triangular regions, or equivalently, that $\Delta f$ is continuous on both triangular regions.

### Completion to the diagonal

Recall that, for any $x_0$, we defined the derivative $f'(x_0)$ as:

$f'(x_0) = \lim_{x \to x_0} \Delta f(x,x_0)$

Due to the symmetry, it can also be defined as:

$f'(x_0) = \lim_{x \to x_0} \Delta f(x_0,x)$

Consider an open interval $I$ and a differentiable function $f$ on $I$. Suppose $f'$ exists on all of $I$. Then, the difference quotient function $\Delta f$ can be extended to the diagonal as the following function $\Delta_c$ (not standard notation, we're just using a slightly different notation from $\Delta f$ to keep track of the distinction):

$\Delta_c f (x_1,x_2) = \left \lbrace \begin{array}{rl} f'(x_1), & x_1 = x_2 \\ \Delta f(x_1,x_2), & x_1 \ne x_2 \\\end{array}\right.$

Note that $\Delta_c$ is a separately continuous function based on the definition of the derivative: it is continuous in each variable holding the other variable's value fixed.

However, $\Delta_c$ need not in general be jointly continuous. Graphically, although it is continuous along horizontal and vertical lines, it need not be continuous along diagonal directions. It turns out that the following holds:

$\Delta_c$ is jointly continuous $\iff$ $f'$ is a continuous function

The forward direction is obvious: if $\Delta_c$ is jointly continuous, the output should vary continuously as we move along the diagonal. The reverse direction follows (in a few steps) from the Lagrange mean value theorem.

For most practical purposes, we simply use the same notation $\Delta$ for $\Delta$ (the difference quotient function proper) and $\Delta_c$ (the difference quotient function completed along the diagonal as the derivative).

### Relation between values at multiple points

The fact that a function of two variables is a difference quotient heavily restricts the permitted types of the function. One obvious relation is that, for points $x_1,x_2,x_3$, we have:

$(x_1 - x_2) \Delta f(x_1,x_2) + (x_2 - x_3) \Delta f(x_2,x_3) + (x_3 - x_1) \Delta f(x_3,x_1) = 0$

This can be rewritten as:

$\Delta f(x_1,x_3) = \frac{x_1 - x_2}{x_1 - x_3} \Delta f(x_1,x_2) + \frac{x_2 - x_3}{x_1 - x_3}\Delta f(x_2,x_3)$

In the case that $x_1 < x_2 < x_3$, we can think of the above as saying that the difference quotient between the two extreme points is a weighted average of the difference quotient between the left and middle point and the difference quotient between the middle and right point, where the weighting is done by the length of the interval.

## Relation with operations on functions

Method for constructing new functions from old In symbols Difference quotient in terms of the old functions and their difference quotients Proof
pointwise sum $f + g$ is the function $x \mapsto f(x) + g(x)$
$f_1 + f_2 + \dots + f_n$ is the function $x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)$
Sum of the difference quotients of the functions being added (the difference quotient of the sum is the sum of the difference quotients)
$\! \Delta f + \Delta g$
$\! \Delta(f_1) + \Delta(f_2) + \dots + \Delta(f_n)$
difference quotient is linear
pointwise difference $f - g$ is the function $x \mapsto f(x) - g(x)$ Difference of the difference quotients, i.e., $\Delta f - \Delta g$ difference quotient is linear
scalar multiple by a constant $af$ is the function $x \mapsto af(x)$ where $a$ is a real number $x \mapsto a \Delta f(x)$ difference quotient is linear
pointwise product $f \cdot g$ (sometimes denoted $fg$) is the function $x \mapsto f(x)g(x)$
$f_1 \cdot f_2 \cdot \dots f_n$ (sometimes denoted $f_1f_2\dots f_n$ is the function $x \mapsto f_1(x)f_2(x) \dots f_n(x)$
For two functions, $\Delta (fg)(x_1,x_2) = f'(x_1)\Delta g(x_1,x_2) + \Delta f(x_1,x_2)g'(x_2)$
For multiple functions, Fill this in later
product rule for divided differences
pointwise quotient $f/g$ is the function $x \mapsto f(x)/g(x)$  ?  ?
composite of two functions $f \circ g$ is the function $x \mapsto f(g(x))$ $(\Delta f \circ (g \times g)) \cdot \Delta g$
Explicitly: $(x_1,x_2) \mapsto \Delta f(g(x_1),g(x_2)) \Delta g(x_1,x_2)$
chain rule for difference quotients

## Reverse-engineering a function from partial information about its difference quotient

### We can only know the function up to additive constants

If two functions differ by a constant, then their corresponding difference quotient functions are identical to each other. This means that even complete knowledge of the difference quotient of a function can only determine the function up to additive constants.

### How much information suffices to determine the function up to additive constants?

The following are true:

• Knowing the restriction of the difference quotient function to any single horizontal or vertical line in the domain suffices. In other words, knowing $\Delta f(x,x_0)$ for all $x$ in the domain and a fixed value of $x_0$ suffices.
• Knowing the restriction of $\Delta$ to the diagonal, i.e., knowing $f'$ (note that this is not quite the restriction of the original difference quotient, but of the difference quotient function completed to the diagonal) suffices to determine $f$ up to additive constants on connected intervals. For domains that have multiple connected components, we determine $f$ up to additive constants on each component, but the constant could differ across the components.
• Knowing the restriction to a line parallel to the diagonal helps determine the function up to addition of a periodic function. Explicitly, if we know $\Delta f(x,x+h)$ for all $x \in \R$, then we know $f$ up to addition of a $h$-periodic function.