Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus → Differential Calculus ↓


Continuity


Continuity is a fundamental concept in calculus that deals with the behavior of a function at a given point. When we say that a function is continuous at a point, it intuitively means that there is no "jump", "break", or "hole" at that point on the graph of the function. However, the formal definition of continuity is more rigorous, involving limits.

The concept of continuity

In simple words, a function f(x) is said to be continuous at a point a if the following three conditions are satisfied:

  1. The function f(x) is defined on a (that is, f(a) exists).
  2. The limit of f(x) exists as x approaches a.
  3. The limit of f(x) as x approaches a is equal to f(a).

In mathematical notation, f(x) is continuous at x = a if:

lim (x → a) f(x) = f(a)

Understanding each situation

1. The function is defined on a

For a function to be continuous at a, we need to ensure that f(a) exists. This means that the function must provide a single, finite value at that point. For example, the function f(x) = 1/x is not defined at x = 0 because division by zero is undefined.

2. The limit of f(x) exists as x → a

The limit of a function as x approaches a exists when the function approaches a particular real number as x approaches a arbitrarily close to a from either side (left and right).

3. The limit is equal to the value a

Finally, for continuity, the limit as x approaches a, denoted by lim (x → a) f(x), must be equal to the value of the function f(a) at that point.

The idea of continuity

To better understand continuity, let's look at some examples with mathematical graphs:

A

continuous function on a.

This graph shows a continuous function. Note that there is no break in the curve at point a. The limit as x approaches a is equal to the function value f(a).

Examples of continuity

Example 1: Polynomial function

Polynomial functions, such as f(x) = x^2 + 2x + 1, are continuous everywhere over their domain, which is all real numbers. This is because they are smooth and have no breaks or holes.

For f(x) = x^2 + 2x + 1, the limit as x approaches any real number is equal to the function value at that point. Therefore, it is continuous everywhere.

Example 2: Rational function

Consider the rational function f(x) = (x^2 - 1) / (x - 1) Before assuming continuity, let's examine this function more closely. The function simplifies to f(x) = x + 1 for all x ≠ 1 At x = 1, the denominator becomes zero, so f(1) is not defined.

lim (x → 1) (x^2 - 1) / (x - 1) = lim (x → 1) x + 1 = 2

Types of discontinuity

A function that is not continuous is called discontinuous. There are several types of discontinuity:

Removable discontinuities

A function has a removable discontinuity at a point if the limit of the function exists at that point but is not equal to the actual value of the function. An example is:

f(x) = (x^2 - 1) / (x - 1) is undefined at x = 1, but can be redefined as a new function that is continuous.

Jump discontinuities

Jump discontinuities occur when the left and right hand limits exist at a point but are not equal to each other. For example, the functions f(x) = 1 for x < 0 and f(x) = 2 for x ≥ 0 have a jump at x = 0.

Infinite discontinuities

A function has an infinite discontinuity at a point if the function approaches infinity as x approaches the point from any direction. For example, f(x) = 1/x has an infinite discontinuity at x = 0.

The discontinuous function exhibits a jump at a.

Piecewise functions and continuity

Piecewise functions composed of several sub-functions defined on different intervals require careful evaluation for continuity. To determine whether a piecewise function is continuous at the boundary point:

  1. Find the left limit as x approaches the limit point from the left.
  2. Find the right-hand limit as x approaches the limit point from the right.
  3. Check if these limits are equal and match the function value at the boundary.

Example: a piece-wise function

Consider a piecewise function:

f(x) = { x + 2 if x < 1 x^2 if x ≥ 1 }
f(x) = { x + 2 if x < 1 x^2 if x ≥ 1 }

To check continuity at x = 1, check the limits:

lim (x → 1-) (x + 2) = 1 + 2 = 3 lim (x → 1+) (x^2) = 1^2 = 1
lim (x → 1-) (x + 2) = 1 + 2 = 3 lim (x → 1+) (x^2) = 1^2 = 1

The left limit is 3, while the right limit is 1, so f(x) is discontinuous at x = 1.

Conclusion

Continuity is an important property in calculus, enabling further analysis and differentiation. Understanding continuity helps illuminate integral and differential calculus concepts and gives insight into complex mathematical modeling. While continuity may seem straightforward, it requires a deeper understanding of limits and function behavior in complex cases such as piecewise functions.


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