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Question

Define g(x)=33f(xy)f(y) dy, for all real x, where
f(t)={1 0t10 elsewhere.
Then

A
g(x) is not continuous everywhere
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B
g(x) is continuous everywhere but differentiable nowhere
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C
g(x) is continuous everywhere and differentiable everywhere except at x=0,1
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D
g(x) is continuous everywhere and differentiable everywhere except at x=0,1,2
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Solution

The correct option is D g(x) is continuous everywhere and differentiable everywhere except at x=0,1,2
g(x)=33f(xy)f(y) dyg(x)=10f(xy) dy
Assuming xy=tdy=dt
So, g(x)=xx1f(t) dt

Case I : x<0
f(t)=0g(x)=0

Case II : 0x<1
g(x)=0x1f(t) dt+x0f(t) dtg(x)=x01 dt=x

Case III : 1x2
g(x)=1x1f(t) dt+x0f(t) dtg(x)=1x11 dt=2x

Case IV : x>2
f(t)=0g(x)=0

g(x)=⎪ ⎪⎪ ⎪0 x<0x 0x<12x 1x20 x>2

Clearly, g(x) is continuous xR and g(x) is differentiable xR{0,1,2}.

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