The Learning Point‎ > ‎Mathematics‎ > ‎

## 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 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!

## ### 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, 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 e-mailed to you at the address you provide.