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Question

Using the definition, show that the function is discontinuous at the point x=0.
f(x)=sinx|x| if x0, and
f(x)=1 if x=0

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Solution

f(x)=sinx|x|x0

Let check L.H.L. limx0sinx|x|

=limx0sinxx [for X<0,|x|=x]

=limx0sinxx

=1

and R.H.L. limx0+sinx|x|

limx0+sinxx [for x>0,|x|=x]

=limx0+sinxx

=1

here L.H.L.R.H.L. so function is discontinuous at x=0

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