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CBSEArts (English Medium) Class 12

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Algebra of Continuous Functions

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Topics

Relations and Functions
- Basics of Relations & Functions
- Types of Relations
- Equivalence Class and Relation
- Types of Functions
- Composition of Functions
- Invertible Functions
Relations and Functions
Algebra
Inverse Trigonometric Functions
- Basics of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
- Overview of Inverse Trigonometric Functions
Matrices
- Concept of Matrices
- Types of Matrices
- Equality of Matrices
- Operations on Matrices> Addition and Subtraction of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Matrix Multiplication
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
- Overview of Matrices
Calculus
Determinants
- Determinant of a Matrix
- Expansion of Determinant
- Area of Triangle using Determinant
- Minors and Co-factors
- Adjoint & Inverse of Matrix
- Applications of Determinants and Matrices
- Overview of Determinants
Vectors and Three-dimensional Geometry
Continuity and Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions
- Derivative of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Overview of Continuity and Differentiability
Linear Programming
Probability
Applications of Derivatives
- Rate of Change of Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
Sets
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Methods of Integration>Integration Using Trigonometric Identities
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Integrals of Some Particular Functions
- Definite Integrals
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Overview of Integrals
Applications of the Integrals
- Area Under Simple Curves
- Overview of Applications of Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Methods of Solving Differential Equations> Variable Separable Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Overview of Differential Equations
Vectors
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
- Types of Vectors in Algebra
- Algebra of Vector Addition
- Multiplication in Vector Algebra
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors
- Overview of Vectors
Three - Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Overview of Three Dimensional Geometry
Linear Programming
- Basic Concepts of Linear Programming
- Linear Programming Problem and Its Mathematical Formulation
- Methods to Solve LPP (Graphical / Corner Point Method)
- Overview of Linear Programming
Probability
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Overview of Probability

Estimated time: 5 minutes

CBSE: Class 12

Introduction

Just as the concept of limits follows specific algebraic rules, the continuity of functions does as well. Because continuity at a point is entirely determined by the limit of the function at that point, the algebra of continuous functions mirrors the algebra of limits.

These rules allow us to easily determine the continuity of complex functions by breaking them down into simpler, known continuous parts.

CBSE: Class 12

Algebra of Continuous Functions

If f and g are two real functions that are continuous at a real number x = c, then the following algebraic combinations are also continuous at x = c:

Addition: f + g
Subtraction: f - g
Multiplication: \[f \cdot g\]
Division: \[\frac{f}{g}\] (provided that \[g(c) \neq 0\])

CBSE: Class 12

Theorem: Composition of Functions

If g is continuous at c, and f is continuous at g(c), then their composite function \[(f \circ g)\], defined as \[(f \circ g)(x) = f(g(x))\], is also continuous at c.

CBSE: Class 12
Maharashtra State Board: Class 12

Key Points: Algebra of Continuous Functions

Operation on continuous functions	Result
f + g	Continuous
f − g	Continuous
fg	Continuous
\[\dfrac{f}{g}\]	Continuous
\[f \circ g\]	Continuous

Video Tutorials

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Series 1
Videos: 1
Duration: 00:06:40
Language: English Series 2
Videos: 1
Duration: 00:14:19
Language: English Series 3
Videos: 2
Duration: 01:07:16
Language: Hindi

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Related QuestionsVIEW ALL [84]

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2kx}{x^2}, \text{ if } & x \neq 0 \\ 8 , \text{ if } & x = 0\end{cases}\] at x = 0

Let \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}, x \neq \frac{\pi}{4} .\] The value which should be assigned to f (x) at \[x = \frac{\pi}{4},\]so that it is continuous everywhere is

Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]

The function f(x) is defined as follows:

\[f\left( x \right) = \begin{cases}x^2 + ax + b , & 0 \leq x < 2 \\ 3x + 2 , & 2 \leq x \leq 4 \\ 2ax + 5b , & 4 < x \leq 8\end{cases}\]

If f is continuous on [0, 8], find the values of a and b.

Discuss the continuity of the function

\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x}{\left| x \right|}, & x \neq 0 \\ 0 , & x = 0\end{array} . \right.\]

Find the points of discontinuity, if any, of the following functions:

\[f\left( x \right) = \begin{cases}x^3 - x^2 + 2x - 2, & \text{ if }x \neq 1 \\ 4 , & \text{ if } x = 1\end{cases}\]

The function \[f\left( x \right) = \begin{cases}x^2 /a , & \text{ if } 0 \leq x < 1 \\ a , & \text{ if } 1 \leq x < \sqrt{2} \\ \frac{2 b^2 - 4b}{x^2}, & \text{ if } \sqrt{2} \leq x < \infty\end{cases}\] is continuous on (0, ∞), then find the most suitable values of a and b.

For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if" x <= 0),(4x+ 1", if" x > 0):}` continuous at x = 0? What about continuity at x = 1?

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