PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Analysis → Real Analysis ↓


Continuity


Continuity is a fundamental concept in real analysis and serves as a cornerstone for more complex discussions in mathematical analysis. At its core, continuity is a way to describe when a function behaves in a "smooth" way without any jumps or breaks. This notion helps mathematicians understand the behavior of functions on certain intervals and provides a way to manipulate and reason about them.

Intuitive idea of continuity

Before diving into formal definitions, let's start with an intuitive idea. Imagine a function like drawing a curve on a piece of paper. A function is continuous if you can draw it on the paper without lifting your pen. If there are any points where you have to lift your pen, then the function is not continuous at those points.

Continuous Not continuous

In the above example, the black line represents a continuous function, while the red line has a jump, which represents a discontinuity.

Formal definition

In formal mathematics, a function f from a set A of real numbers to the real numbers is said to be continuous at a point c in A if for every positive number ε, no matter how small, there exists a positive number δ such that whenever x lies within δ of c, f(x) lies within ε of f(c).

For all ε > 0, there exists a δ > 0 such that if |x - c| < δ, then |f(x) - f(c)| < ε.

This definition might seem a bit complicated, but it will make more sense as we explore examples and dive deeper into its implications.

Understanding ε-δ definition

ε-δ definition of continuity formalizes the intuition that we should be able to make the output of a function as close as we want to a given point by taking inputs close enough to it. Let's understand what this means in simple terms.

- ε (epsilon): indicates how close we want the function value f(x) to be to f(c).

- δ (delta): This indicates the range around c within which we can choose our x.

Example with linear function

Consider the function f(x) = 2x + 3 We will show that this function is continuous everywhere on the real line. Take any point c and verify ε-δ condition.

|f(x) - f(c)| = |(2x + 3) - (2c + 3)| = |2x + 3 - 2c - 3| = |2x - 2c| = 2|x - c|

We want 2|x - c| < ε. This implies |x - c| < ε/2. So, for every ε > 0, we can choose δ = ε/2, which easily shows that the function is continuous everywhere.

Visualization of the epsilon-delta state

f(c) f(x) in (f(c)-ε, f(c)+ε) (c−δ, c+δ) in x

More text examples

To understand this concept further, let us consider some more functions.

Example: Quadratic function

Consider f(x) = x^2. We can show continuity at any point c by examining how changes to x affect f(x). Specifically:

|f(x) - f(c)| = |x^2 - c^2| = |(x - c)(x + c)|

Let |x - c| < 1, then |x + c| < 2|c| + 1 thus:

|f(x) - f(c)| = |x - c||x + c| < δ(2|c| + 1)

Set δ = ε/(2|c| + 1) This option supports continuity of the function.

Example: sine function

Consider f(x) = sin(x). The sine function is assumed to be continuous everywhere. Let's observe a point c using the properties of sine.

|f(x) - f(c)| = |sin(x) - sin(c)|

Using the identity for sine difference:

|sin(x) - sin(c)| = 2|cos((x+c)/2)sin((x-c)/2)|

Since the cosine and sine functions are bounded by 1, continuity is immediately achieved by choosing an appropriate δ.

Continuing expansions and their significance

Continuity also allows the extension of a function. A function defined on a part of the real line can be extended to all parts of the real line in a continuous way. Understanding when and how we can extend a function continuously is valuable in both theory and applications.

Example: function defined on a subgroup

Consider the function

f(x) = 1/x

Defined on the set (0, ∞). This function cannot be extended to x = 0 without discontinuity, since f(x) diverges as it approaches zero.

Properties of continuous functions

Continuous functions exhibit special properties that are useful in mathematics, such as:

  • Intermediate Value Property: A function f that is continuous on the interval [a, b] attains every value between f(a) and f(b).
  • Extreme Value Theorem: A function that is continuous on a closed interval attains a maximum and a minimum value.
  • Preservation of compactness: the image of a compact set under a continuous function is compact.

Compact groups and continuity

In dealing with functions on the real numbers, dense subsets play an important role. A set is dense in an interval if between any two numbers in the set, there exist numbers from the set. Continuous functions exhibit the important property that if they are equal to zero on a dense set, they must be zero everywhere.

This property has important implications in the construction of functions and in the extension of results from finite sets to arbitrary sets.

Conclusion

Continuity, while theoretically quite simple, has profound implications for real analysis and advanced mathematical studies. It provides insights into the structure and behavior of functions that can have far-reaching applications in calculus, differential equations, and even advanced fields such as topology and functional analysis. This concept determines much of the behavior of a function, giving mathematicians a toolset for exploring the infinite realm smoothly and predictably.


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Continuity

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