Outline of Contents in this tutorial (There is a PDF Document after this outline)
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 ALevel; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE/AIEEE Anyone else who needs this Tutorial as a reference!
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 saidto 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 approachesa(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 writelim f (x) = b_{1} x→a− ..then b_{1} 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, b_{1} = b_{2} = 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, after understanding the basics)
MCQ Quiz on the basics of Functions, Limits, Continuity Companion MCQ Quiz for Functions, Limits, Continuity test how much you know about the topic. Your score will be emailed to you at the address you provide.
MCQ Quiz Functions, Limits, ContinuityGoogle Spreadsheet Form
Our Set of Calculus Tutorials Quick and introductory definitions related to Funtions, Limits and Continuity Functions, Limits and Continuity  Solved Problem Set I  The Domain, Range, Plots and Graphs of Functions; L'Hospital's 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 Functions, Limits and Continuity  Solved Problem Set III  Continuity and Intermediate Value Theorems Introductory concepts and definitions related to Differentiation  Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc Differential Calculus  Solved Problem Set I  Common Exponential, Log , trigonometric and polynomial functions Differential Calculus  Solved Problem Set II  Derivability and continuity of functins  Change of Indepndent Variables  Finding Nth Derivatives 
Differential Calculus  Solved Problems Set III Maximia, Minima, Extreme Values, Rolle's Theorem Differential Calculus  Solved Problems Set IV  Points of Inflexion, Radius of Curvature, Curve Sketching Differential Calculus  Solved Problems Set V  Curve Sketching, Parametric Curves 
Introducing Integral Calculus  Definite and Indefinite Integrals  using Substitution , Integration By Parts, ILATE rule Integral Calculus  Solved Problems Set I  Basic examples of polynomials and trigonometric functions, area under curves Integral Calculus  Solved Problems Set II  More integrals, functions involving trigonometric and inverse trigonometric ratios Integral Calculus  Solved Problems Set III  Reduction Formulas, Using Partial FractionsI Integral Calculus  Solved Problems Set IV  More of integration using partial fractions, more complex substitutions and transformations Integral Calculus  Solved Problems Set V Integration as a summation of a series Introduction to Differential Equations and Solved Problems  Set I  Order and Degree, Linear and NonLinear Differential Equations, Homogeneous Equations, Integrating Factor Differential Equations  Solved Problems  Set II  D operator, auxillary equation, General Solution Differential Equations  Solved Problems  Set III  More Differential Equations Differential Equations  Solved Problems  Set IV

