Q. 95.0( 4 Votes )

Discuss the continuity of the function

Answer :

Basic Idea:


A real function f is said to be continuous at x = c, where c is any point in the domain of f if :


where h is a very small ‘+ve’ no.


i.e. left hand limit as x c (LHL) = right hand limit as x c (RHL) = value of function at x = c.


This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :



Here we have,


…….equation 1


Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain ( domain = set of numbers for which f is defined )


Function is changing its nature (or expression) at x = 2, So we need to check its continuity at x = 2 first.


LHL = = = [using eqn 1]


RHL = [using eqn 1]


f(2) =


[using eqn 1]


Clearly, LHL = RHL = f(2)


function is continuous at x = 2


Let c be any real number such that c > 2


f(c) = [using eqn 1]


And,


Thus,


f(x) is continuous everywhere for x > 2.


Let m be any real number such that m < 2


f(m) = [using eqn 1]


And,


Thus,


f(x) is continuous everywhere for x < 2.


Hence, We can conclude by stating that f(x) is continuous for all Real numbers


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