# Differentiability and derivative

Continuity of functions (and maps) means that they can be nicely approximated by constant functions (maps) in a sufficiently small neighborhood of each point. Yet the constant maps (easy to understand as they are) are not the only “simple” maps.

## Linear maps

Linear maps naturally live on *vector spaces*, sets equipped with a special structure. Recall that is algebraically a *field*: real numbers cane be added, subtracted between themselves and the ratio is well defined for .

**Definition.** A set is said to be a vector space (over ), if the operations of addition/subtraction and multiplication by constant are defined on it and obey the obvious rules of commutativity, associativity and distributivity. Some people prefer to call vector spaces *linear spaces*: the two terms are identical.

**Warning.** There is no “natural” multiplication !

**Examples.**

- The field itself. If we want to stress that it is considered as a vector space, we write .
- The set of tuples is the Euclidean -space. For it can be identified with the “geometric” plane and space, using coordinates.
- The set of all polynomials of bounded degree with real coefficients.
- The set of all polynomials without any control over the degree.
- The set of all continuous functions on the segment .

**Warning.** The two last examples are special: the corresponding spaces are not finite-dimensional (we did not have time to discuss what is the dimension of a linear space in general…)

Let be two (different or identical) vector spaces and is a function (map) between them.

**Definition. ** The map $f$ is linear, if it preserves the operations on vectors, i.e., .

Sometimes we will use the notation .

**Obvious properties of linearity.**

- (Note: the two zeros may lie in different spaces!)
- For any two given spaces the linear maps between them can be added and multiplied by constants in a natural way! If , then we define for any (define yourselves). The result will be again a linear map between the same spaces.
- If and , then the composition is well defined and again linear.

**Examples.**

- Any linear map has the form (do you understand why the notations are used?)
- Any linear map has the form for some numbers . Argue that all such maps form a linear space isomorphic to back again.
- Explain how linear maps from to can be recorded using -matrices. How the composition of linear maps is related to the multiplication of matrices?

The first example shows that linear maps of to itself are “labeled” by real numbers (“*multiplicators*“). Composition of linear maps corresponds to multiplication of the corresponding multiplicators (whence the name). A linear 1-dim map is invertible if and only if the multiplicator is nonzero.

**Corollary.** *Invertible linear maps constitute a commutative group (by composition) isomorphic to the multiplicative group . *

## Shifts

Maps of the form for a fixed vector (the domain and source coincide!) are called *shifts* (a.k.a. translations). **Warning:** The shifts are not linear unless ! Composition of two shifts is again a shift.

**Exercise.**

*Prove that all translations form a commutative group (by composition) isomorphic to the space itself.* (Hint: this is a tautological statement).

## Affine maps

**Definition.**

A map between two vector spaces is called *affine*, if it is a composition of a linear map and translations.

**Example.**

Any affine map has the form for some . Sometimes it is more convenient to write the map under the form : this is possible for any point . Note that the composition of affine maps in dimension 1 is not commutative anymore.

**Key computation.** Assume you are given a map in the sense that you can evaluate it at any point . Suppose an oracle tells you that this map is affine. How can you restore the explicit formula for ?

Obviously, . To find , we have to plug into it any point and the corresponding value . Given that , we have for any choice of .

The expression for a non-affine function is in general not-constant and depends on the choice of the point .

**Definition.** A function is called differentiable at the point , if the above expression for , albeit non-constant, has a limit as , where is a function which tends to zero. The number is called the **derivative** of at the point and denoted by (and also by half a dozen of other symbols: , …).

Existence of the limit means that near the point the function admits a reasonable approximation by an affine function : , i.e., the “non-affine part” is small not just by itself, but also relative to small difference .

# Differentiability and algebraic operations

See the notes and their earlier version.

The only non-obvious moment is differentiability of the product: the product (unlike the composition) of affine functions is not affine anymore, but is immediately differentiable:

, but the quadratic term is vanishing relative to , so the entire sum is differentiable.

**Exercise.** Derive the Leibniz rule for the derivative of the product.

# Derivative and the local study of functions

Affine functions have no (strong) maxima or minima, unless restricted on finite segments. Yet absence of the extremum is a strong property which descends from the affine approximation to the original function. Details here and here.