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## Calculus - Introductory Definitions related to Functions, Limits and Continuity - Outline of Contents:

### Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits;  introducing the L'Hospital rule, intermediate and extreme value theorems.

### Domain and Range of a Function: Let A and B be any two sets and let f denotes a rule which associates to each member of A a member of B. We say that f is a function from A into B. Also A is said

to be Domain of this function. If x denotes a member of the set A, then the member of the set B, which the function f associates to x ∈ A, is denoted by f (x) called the value of the function f for x or at x. The function may be described as x → f (x), or y = f (x) where x ∈ A and y ∈ B.
Range of f = {f (x) : x ∈ A}.

### • Continuity: A function f is continuous at a point b in the domain of f if and only if for each positive real number ε such that for each x in the domain of f |x − b| < δ =⇒ |f (x) − f (b)| < ε.

A function f is continuous at a point b if and only if
1. b is in the domain of f
2. lim x→b f (x) exists
3. lim x→b f (x) = f (b)

### • Limit: Let the function y = f (x) be deﬁned in a certain neighbourhood of a point a or at certain points of this neighbourhood. The function y = f (x) approaches the limit b(y → b) as x approaches

a(x → a), if for every positive number ε, no matter how small, it is possible to indicate a positive number δ such that for all x, diﬀerent from a and satisfying |x − a| < δ, we have |f (x) − b| < ε
If b is the limit of the function f (x) as x → a, we write
lim f (x) = b
x→a

### • Left hand and Right Hand Limits: If f (x) approaches the limit b1 as x takes on only values less than a we write

lim f (x) = b1
x→a−
..then b1 is called the limit on the left at the point a of the function. If x takes on only values greater than a, we write lim f (x) = b2
x→a+
.. then b2 is called the limit on the right at the point a of the function. If the limit on the right and the limit on the left exist and equal, that is, b1 = b2 = b, then b will be the limit of f (x) at point a.

These are some of the rules which we will we introduce in this tutorial- Sum Rule, Product Rule, Quotient Rule, Constant Multiple Rule, Power Rule, Limit Of an Exponential Function, Limit of a Logarithm of a Function, L'Hospital Theorem, Common limits such as six/x, tan x/ x

## This tutorial also introduces the Continuity Theorems, Extreme Value Theorem, Intermediate Value Theorem:

1. Let the function f (x) be continuous at x = a and let c be a constant. Then the function cf (x) is also continuous at x = a.
2. Let the functions f (x) and g(x) be continuous at x = a. Then the sum of the functions f (x) + g(x) is also continuous at x = a.
3. Let the functions f (x) and g(x) be continuous at x = a. Then the product of the functions f (x)g(x) is also continuous at x = a.
4. Let the functions f (x) and g(x) be continuous at x = a. Then the quotient of the functions f (x) is also continuous at x = a.
5. Let f (x) be diﬀerentiable at the point x = a. Then the function f (x) is continuous at that point. But the converse is not true.
6. Extreme Value Theorem: If f (x) is continuous on the closed, bounded interval [a, b], then it is bounded above and below in that interval. That is there exists numbers m and M such that m ≤ f (x) ≤ M , for every x in [a, b]
7. Intermediate Value Theorem: Let f (x) be continuous on the closed, bounded interval [a, b]. Then if c is any number between f (a) and f (b), there is a number x0 such that
f(c) =

## Complete Tutorial Document (Also check out the MCQ Quiz below this):

### MCQ Quiz on Functions, Limits, Continuity (The Basics)

Companion MCQ Quiz for Functions, Limits, Continuity- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

### In Case you'd like to take a look at some of our other tutorials related to Single Variable Calculus :

 Quick and introductory definitions related to Funtions, Limits and Continuity - Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits;  defining the L'Hospital rule, intermediate and extreme value theorems. Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions;  L'Hospital's Rule- -  Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule. Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x   - More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial). Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem. Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc - Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions - Examples and solved problems - differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability . Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives - Examples and solved problems - related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the N-th derivative of functions Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values,  Rolle's Theorem - Examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching -  Examples and solved problems - Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  - Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions.  Definite integration and properties of definite integrals; Integration by  substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration. Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves - Examples and solved problems - elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves. Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios - Examples and solved problems - integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios. Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI- Examples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite integrals Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations - Examples and solved problems - More of integrals involving partial fractions, more complex substitutions and transformations Integral Calculus - Solved Problems Set V- Integration as a summation of a series - Examples and solved problems - More complex examples of integration, examples of integration as the limit of a summation of a series Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor -  Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution - Examples and solved problems - Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI ) Differential Equations - Solved Problems - Set III - More Differential Equations - More complex cases of differential equations. Differential Equations - Solved Problems - Set IV - Still more differential equations.