Q. 105.0( 3 Votes )

Find which of the functions is continuous or discontinuous at the indicated points:

f(x) = |x| + |x – 1| at x = 1

Answer :

Given,


f(x) = |x| + |x – 1| …(1)


We need to check its continuity at x = 1


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 according to above theory-


f(x) is continuous at x = 1 if -



Clearly,


LHL = {using eqn 1}


LHL =


h > 0 as defined above and h0


|-h| = h


And (1 – h) > 0


|1 – h| = 1 - h


LHL =


LHL = 1 …(2)


Similarly we proceed for RHL-


RHL = {using eqn 1}


RHL =


h > 0 as defined above and h0


|h| = h


And (1 + h) > 0


|1 + h| = 1 + h


RHL =


RHL = 1 + 2(0) = 1 …(3)


And,


f(1) = |1|+|1-1| = 1 {using eqn 1} …(4)


Clearly from equation 2 , 3 and 4 we can say that



f(x) is continuous at x = 1


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