Q. 25.0( 3 Votes )

Discuss the conti

Answer :

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 the limit from class 11 we can summarise it as


A function is continuous at x = c if :



Here we have,


…….equation 1


The 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 = 0, So we need to check its continuity at x = 0 first.


NOTE:


LHL = = =


[using eqn 1 and idea of mod fn]


RHL =


[using eqn 1 and idea of mod fn]


f(0) = 0


[using eqn 1]


Clearly, LHL ≠ RHL ≠ f(0)


function is discontinuous at x = 0


Let c be any real number such that c > 0


f(c) =


[using eqn 1]


And,


Thus,


f(x) is continuous everywhere for x > 0.


Let c be any real number such that c < 0


f(c) =


[using eqn 1 and idea of mod fn]


And,


Thus,


f(x) is continuous everywhere for x < 0.


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


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