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Discuss the continuity of the functions at the points shown against them . If a function is discontinuous , determine whether the discontinuity is removable . In this case , redefine the function , so that it becomes continuous : `{:(f(x)=xsin(1/x)" , for "x ne0),(=0" , for "x=0):}}` at `x=0`

Text Solution

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`f(0)=0" "…"(Given")…(1)`
`underset(x to0)limf(x)=underset(x to0)lim x sin1/x`
Since `-1lesin""1/xle1" "as x ne0`
`therefore-xlex sin 1/xlex`
`thereforeunderset(xto0)lim(-x)leunderset(x to0)lim x sin1/x leunderset(xto0)lim(x)`
`therefore 0leunderset(xto0)lim f(x)le0`
`thereforeunderset(x to0)limf(x)=0" "...(2)`
From (1) and (2) , `underset(x to 0) lim f(x)=f(0)`
`therefore f` is continuous at `x=0.`
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