(5)

f(x) is continuous at x=0. Check with LHL=RHL = value of the function.

f(x) is clearly differentiable. Check with left hand differentiability and right hand differentiability at x=0 to see that f(x) doesn't have a "kink" at x=0.

f'(x)= 2xsin(1/x)-cos(1/x) when x ≠ 0

= 0 when x=0

now we need to check the continuity of g(x)=f'(x)

the limit of g(x) is definitely ≠ 0 (infact it doesn't even exist) when x tends to zero. hence g(x) which is nothing but f'(x) is discontinuous at x=0.

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"You don't have to believe in God, but you should believe in The Book." -Paul Erdős