Formal Definition of a Function

This document provides a comprehensive overview of fundamental concepts in mathematical analysis, focusing on functions, limits, continuity, and differentiability.

Functions: Basic Definitions and Properties

A function is a relation where each input maps to exactly one output.

Key Terminology

• Digital function on a set E: A process that associates at most one element of (image ) to each element of . is the preimage.

• Domain (): The set of elements in that have an image under .

• Set of Departure (E): The input set.

• Set of Arrival (R): The target output set.

• Real function of a real variable: If .

• Image (Range, or ): The set of all actual output values .

Graphical Representation

• The graph of a function is the set of ordered pairs , where .
  

• These ordered pairs represent points with in a plane.

Characteristics of Functions

• Even function: . Graph is symmetric with respect to the y-axis.

• Odd function:

mark>. Graph is symmetric with respect to the origin.

• Periodic function: for all , where is the period.

  • The fundamental period is the least positive value of .

  • Examples: and are -periodic; is -periodic.

Monotonicity (Variations)

Let .

• Increasing: .

• Strictly increasing: .

• Decreasing: .

• Strictly decreasing: .

• Monotonic: Either increasing or decreasing.

• Strictly monotonic: Either strictly increasing or strictly decreasing.

Boundedness

Let .

• Bounded from above: such that for all .

• Bounded from below: such that for all .

• Bounded: Bounded from both above and below; such that for all .

  • Supremum (): Least upper bound of .

  • Infinimum (): Greatest lower bound of .

• Unbounded: Not bounded.

Operations on Functions

Let be real functions, .

• Sum ():

  • Domain: .

  • Value: .

• Scalar Product ():

  • Domain: .

  • Value: .

• Product ():

  • Domain: .

  • Value: .

• Quotient ():

  • Domain: .

  • Value: .

• Composition (): If and with .

  • Value: . Order matters!

Comparison: if . if .

Types of Functions

• Injective (One-to-one):

  • .

  • Equivalently: .

  • A strictly monotone function is injective.

• Surjective (Onto):

  • .

  • The range equals the codomain.

• Bijective: Both injective and surjective.

Limits of Functions

The limit of a function as approaches (denoted ) means values get arbitrarily close to when is close to (but not necessarily equal to) .

Formal Definition (Epsilon-Delta)

• if such that for all , if , then .

• A neighborhood of a real number is a set that contains an open interval containing .

• The limit, if it exists, is unique.

One-Sided Limits

• Right-hand limit: if such that for all , if , then .

• Left-hand limit: if such that for all , if , then .

• if and only if .

Infinite Limits and Limits at Infinity

• : such that if , then .

• : such that if , then .

• : such that if , then .

• : such that if , then .

• $\lim_{x \to +\</mark></p></li></ul><p></p><p>infty} f(x) = +\infty$, etc. (various combinations of infinite limits).

  Sequential Criterion for Limits

  • if and only if for every sequence in such that and , the sequence converges to .

  Properties of Limits (Limit Laws)

  If and :

  • .

  • .

  • .

  • (if ).

  • .

  • Composition Rule: If and (g is continuous at y0), then .

  Comparison Theorems

  • If and limits exist, then .

  • Squeeze Theorem: If and , then .

  Indeterminate Forms

  • Occur when limit properties cannot be directly applied due to expressions like .

  • Require further analysis (e.g., L'Hôpital's Rule).

  Continuity of Functions

  A function is continuous if its graph can be drawn without lifting the pen.

  Definition

  • is continuous at if such that for all , if , then .

  • This is equivalent to .

  • If is not continuous at , it is discontinuous at .

  Types of Continuity

  • Continuity on the right at : .

  • Continuity on the left at : .

  • A function is continuous at if and only if it is continuous on the right and on the left at .

  Continuity on Intervals

  • Continuous on an open interval : Continuous at every point in .

  • Continuous on a closed interval : Continuous on , continuous from the right at , and continuous from the left at .

  Sequential Criterion for Continuity

  • is continuous at if and only if for every sequence in converging to , the sequence converges to .

  Properties of Continuous Functions

  If are continuous at :

  • is continuous at ().

  • is continuous at .

  • is continuous at .

  • is continuous at (if ).

  • Polynomials are continuous on .

  • Rational functions are continuous wherever they are defined (denominator non-zero).

  • Composition of Continuous Functions: If is continuous at and is continuous at , then is continuous at .

  Continuous Extension

  If is undefined but exists and is finite, can be extended to a continuous function at by defining . $F(x) = \begin{cases} f(x) &amp; \text{if } x \in D_f \\ \lim_{x \to c} f(x) &amp; \text{if } x = c \end{cases}" data-type="inline-math"></span><mark>$

  Uniform Continuity

  • is uniformly continuous on if such that for all , if , then .

    • The depends only on , not on the specific point .

    • If uniformly continuous, then continuous. The converse is not always true.

  • Lipschitz function: for all and some constant .

    • A Lipschitz function is uniformly continuous.

  • Uniform Continuity Theorem: If is continuous on a closed bounded interval , then is uniformly continuous on .

  Key Theorems on Continuous Functions on Closed Bounded Intervals

  • Boundedness Theorem: If is continuous on a closed bounded interval , then is bounded on (i.e., for all ).

  • Maximum-Minimum Theorem: If is continuous on a closed bounded interval , then attains its absolute maximum and absolute minimum on . That is, there exist such that and .

  • Location of Roots Theorem: If is continuous on

  and and have opposite signs ( or ), then there exists a number such that .

  • Intermediate Value Theorem (IVT): If is continuous on an interval , and is any value between and for , then there exists a point between and such that .

    • Image of an interval: If is an interval and is continuous on , then is also an interval.

  • Fixed Point Theorem: If is continuous, then there exists such that .

  • Continuous Inverse Theorem: If is strictly monotone and continuous on an interval , then its inverse function exists, is strictly monotone, and continuous on .

  Differentiability of Functions

  Differentiability indicates the existence of a well-defined tangent line at each point of the function's graph.

  Definition

  • is differentiable at if the limit exists and is finite.

  • This limit is the derivative of at , denoted , , or .

  • Alternatively: .

  • The equation of the tangent line to at is .

  One-Sided Derivatives

  • Left derivative: .

  • Right derivative: .

  • is differentiable at if and only if and exist and are equal.

  Differentiability Implies Continuity

  • If has a derivative at , then is continuous at .

    • The converse is false: A function can be continuous but not differentiable (e.g., at ).

  Differentiability on an Interval

  • A function is differentiable on an interval if it is differentiable at every point of

  mark>.

  Rules of Differentiation

  If are differentiable at :

  • Scalar Multiple Rule: .

  • Sum Rule: .

  • Product Rule: .

  • Quotient Rule: (if ).

  • Chain Rule: If is differentiable at and is differentiable at , then .

    • Specific applications:

      • .

      • .

      • (for ).

  • Inverse Function Theorem: If is bijective, continuous, and differentiable at with , then is differentiable at , and .

  Higher Order Derivatives

  • Second derivative: .

  • n-th derivative: . Notation: or .

  • Leibniz Rule (for product of functions): .

  Classes of Functions ()

  • or : Functions continuous on .

  • : Functions differentiable times on with the -th derivative being continuous on .

    • .

  • : Functions infinitely differentiable on (i.e., have derivatives of all orders).

  Extrema of Functions

  • Local (Relative) maximum at : for all in a neighborhood of .

  • Local (Relative) minimum at : for all in a neighborhood of .

  • Local (Relative) extremum: A local maximum or minimum.

  • Global (Absolute) maximum at : for all .

  • Global (Absolute) minimum at

  : for all .

  • Global (Absolute) extremum: A global maximum or minimum.

  Theorems Related to Derivatives

  • Interior Extremum Theorem: If has a relative extremum at an interior point of an interval , and exists, then .

  • Rolle's Theorem: If is continuous on , differentiable on , and , then there exists at least one such that .

  • Mean Value Theorem (MVT): If is continuous on and differentiable on , then there exists at least one such that .

    • Geometric interpretation: There's a point where the tangent line is parallel to the secant line connecting and .

    • Applications:

      • If on , then is increasing on .

      • If on , then is decreasing on .

  • Cauchy Mean Value Theorem (Generalized MVT): If are continuous on and differentiable on , with on , then there exists such that .

  • L'Hôpital's Rule: For indeterminate forms or , if exists (and other conditions met), then .