The concept of limits leads to define and describe continuity and derivative of the function. The continuity of a function has practical as well as theoretical importance. We plot graphs by taking the values generated in the laboratory or collected in the field. We connect the plotted points with a smooth and unbroken curve (continuous curve). This continuous curve helps as to estimate the values at the places where we haven't measured. It was developed by Isaac Newton and Leibnitz.
Here in this chapter, we will study some standard functions, their graphs, concept of limits and discuss about the continuity of the functions. Throughout this chapter, we denote R as the set of real numbers.
Real Functions and their Graphs
Real Function: A real valued function f : A to B or simply a real function 'f ' is a rule which associates to each possible real number x
A, a unique real number f(x)
B, when A and B are subsets of R, the set of real numbers.
Operation on Real Functions
The following are the Operation on Real Functions: Sum Function, Difference Function, Product Function, Quotient Function, Scalar Multiplication Function, Composite Functions, Inverse Functions.
Limits
Left Hand Limit: Let f(x) tend to a limit l1 as x tends to a through values less than 'a', then l1 is called the left hand limit.
Right Hand Limit: Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit.
We say that limit of f(x) exists at x = a, if l1 and l2 are both finite and equal.
Limits (Contd....)
Limits of Trigonometric Functions and Sandwich Theorem:
for all x in some open interval containing c and suppose
Limits (Contd....)
Limits at infinity:
If x is a variable such that it can take any real value how much ever 
The two important properties of these one-sided limits that
i) If the left hand limit and right hand limit of a function at a point exists, but are not equal, then we conclude that the limit at that point does not exist.
ii) If LHL and RHL of a function at a point (say a) exist and they are equal, we conclude that limit at that point exists and we write
Neighborhood of a Point
Let a be a real number. Then for a positive real number δ>0 the interval (a- δ, a+ δ) is called the
neighborhood of a. The interval (a- δ, a) is called a left hand neighborhood of a, and (a, a+ δ) is a
right hand neighborhood of a. If x є(a, a+ δ) we say x approaches a from right and we write x->a+.If
x є (a- δ, a) we say x approaches a from left and write x-> a-.
Algebra of Limits
If f and g are two functions defined over same domain D, then we have certain set of identities
which can be used for solving limits problems with variables like algebraic expressions,
Continuity at a Point
1. We say that f(x) is continuous if f(x) is continuous at every point in its domain.
2. If f and g are two continuous functions then f + g, f - g, fg are continuous functions.
Problems on Limits
Here is a list of problems solved using the identities of limits, standard limits, limits theorem
explained above....
Summary
Every polynomial is continuous. Every rational function is continuous.
Conclusion
In this chapter, we have studied various types of functions and their graphs. The use of graphs also facilitate the study of domain and range of functions.
