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Question

Let f,g and h be three functions defined as
f(x)={xfor|x|11for|x|>1,

g(x)=cos(π2x)for|x|1|x1|for|x|>1

and h(x)=|x|1loga|x|for|x|1lnafor|x|=1 for a>0 and a1

If l,m,n denote the number of points of discontinuity of the functions f,g and h in their domain respectively, over R, then (l,m,n) is

A
(0,0,0)
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B
(1,1,1)
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C
(2,2,2)
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D
(1,1,0)
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Solution

The correct option is D (1,1,0)
f(x)={xfor1x11for x>1 or x<1

Clearly, f(x) is discontinuous at x=1

g(x)=⎪ ⎪⎪ ⎪cosπx2 for1x1x1 for x>11x for x<1

Clearly, g(x) is discontinuous at x=1

h(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪x1logaxfor x>0,x1x1loga(x)for x<0,x1lnafor x=1 or x=1

x=0 is not in the domain of h(x)
limx1±h(x)=(x1logax) 00 form
Applying L'Hospital rule
limx1±h(x)=limx1±xlna=lna=h(1)
h(x) is continuous at x=1

limx1±h(x)=(x1loga(x)) 00 form
Applying L'Hospital rule
limx1±h(x)=limx1±(xlna)=lna=h(1)
h(x) is continuous at x=1
Clearly, h(x) is continuous for all x in its domain.
Hence, there is no point of discontinuity.

Now, (l,m,n)=(1,1,0)

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