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please explain me the concept of continuity of a function and meaning of proving the equality of left hand limit, right hand limit and the value of the function at a particular point for establishing continuity of a function.

please explain me the concept of continuity of a function and meaning of proving the equality of left hand limit, right hand limit and the value of the function at a particular point for establishing continuity of a function.

Grade:11

1 Answers

Hareesh Mahadevan
23 Points
14 years ago

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 xA, 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 ?(a, a+ δ) we say x approaches a from right and we write x->a+.If

x ? (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

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