- 1 Definition
- 2 Definition as a function
- 3 Related notions
- 4 Properties of the difference quotient function
- 5 Relation with operations on functions
- 6 Reverse-engineering a function from partial information about its difference quotient
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 is a function defined on some subset of the reals and are distinct elements in the domain of , then the difference quotient of between and , denoted , is defined as:
Note that the definition is symmetric in and , i.e., we have:
In symbols, if is a function defined on some subset of the reals and are distinct elements in the domain of , then the difference quotient of between and is defined as the slope of the line segment joining the points and , both of which are part of the graph of .
Definition as a function
Consider a function with domain a subset of . The difference quotient, denoted , is a function defined on where is the diagonal subset . In other words, the difference quotient is defined on the set of ordered pairs . It is defined as:
The function is symmetric, i.e., . Therefore, we can only think of it as a function on unordered pairs, i.e., we can view as a function on the set of unordered pairs of distinct elements of .
- 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
The difference quotient function is symmetric: for a function on a subset of , and for distinct points of , we have:
Explicitly, given an interval , and a continuous function on , the domain of is a union of two triangular regions in , 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 is continuous at every point in both triangular regions, or equivalently, that is continuous on both triangular regions.
Completion to the diagonal
Recall that, for any , we defined the derivative as:
Due to the symmetry, it can also be defined as:
Consider an open interval and a differentiable function on . Suppose exists on all of . Then, the difference quotient function can be extended to the diagonal as the following function (not standard notation, we're just using a slightly different notation from to keep track of the distinction):
Note that 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, 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:
is jointly continuous is a continuous function
The forward direction is obvious: if 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 for (the difference quotient function proper) and (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 , we have:
This can be rewritten as:
In the case that , 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|| is the function
is the function
| Sum of the difference quotients of the functions being added (the difference quotient of the sum is the sum of the difference quotients)
||difference quotient is linear|
|pointwise difference||is the function||Difference of the difference quotients, i.e.,||difference quotient is linear|
|scalar multiple by a constant||is the function where is a real number||difference quotient is linear|
|pointwise product|| (sometimes denoted ) is the function
(sometimes denoted is the function
| For two functions,
For multiple functions, Fill this in later
|product rule for divided differences|
|pointwise quotient||is the function||?||?|
|composite of two functions||is the function||
|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 for all in the domain and a fixed value of suffices.
- Knowing the restriction of to the diagonal, i.e., knowing (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 up to additive constants on connected intervals. For domains that have multiple connected components, we determine 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 for all , then we know up to addition of a -periodic function.