Find that the function is continuous or discontinuous at the indicated point `f(x) = {{:(|x-a|sin\ (1)/(x-a),if x ne a),(0, if x =a):}` at `x = a`

Answer

N/a

Explanation

We have, `f(x) = {{:(|x|sin'(1)/(x-a),if x ne 0),(0, if x =a):}` at `x = a`
At `LHL = underset(xrarra^(-))(lim)|x-a|sin'(1)/(x-a)`
`=underset(hrarr0)(lim) |a-h-a| sin((h)/(a-h-a))`
`= underset(hrarr0)(lim)-hsin(1/h) , [:' sin(-theta)=-sintheta]`
`= 0 xx` [an oscillating number between `-1` and 1] = 0
`RHL = underset (xrarra^(+))(lim)|x-a|sin((1)/(x-a))`
`= underset(hrarr0)lim|a+h-a|sin((1)/(a+h-a))= underset(hrarr0)limh sin '1/h`
`= 0xx` [ an oscillating number between ` -1` and 1] = 0
and `f(a) = 0`
`:. LHL = RHL = f(a)`
So, `f(x)` is continous at `x = a`.