Q. 155.0( 2 Votes )

Prove that the function f defined by



remains discontinuous at x=0, regardless the choice of k.

Answer :

Given,


…(1)


We need to prove that f(x) is discontinuous at x = 0 irrespective of the value of k.


A function f(x) is said to be continuous at x = c if,


Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).


Mathematically we can represent it as-



Where h is a very small number very close to 0 (h0)


Now, We need to prove that f(x) is discontinuous at x = 0 irrespective of the value of k


If we show that,



Then there will not be involvement of k in the equation & we can easily prove it.


So let’s take LHL first –


LHL =


LHL =


LHL =


h > 0 as defined in theory above.


|-h| = h


LHL =


LHL =


LHL = …(2)


Now Let’s find RHL,


RHL =


RHL =


RHL =


h > 0 as defined in theory above.


|h| = h


RHL =


RHL =


RHL = …(3)


Clearly form equation 2 and 3,we get


LHL ≠ RHL


Hence,


f(x) is discontinuous at x = 0 irrespective of the value of k.


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