Prove that f is differentiable on |R
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f(x) = x^4 ( 2 + sin(1/x) ) for x not 0, g(0)=0.
Then, for h not zero:
| f(0+h) - f(0) | / h = | 2h^4 + h^4 sin(1/h) | / h
= ± |h|^3 |2+sin(1/h)|
≤ ± |h|^3 (2 + |sin(1/h)| )
≤ ± 3 |h|^3
which obviously goes to 0 as h->0.
So f is differentiable at 0.
Then, for h not zero:
| f(0+h) - f(0) | / h = | 2h^4 + h^4 sin(1/h) | / h
= ± |h|^3 |2+sin(1/h)|
≤ ± |h|^3 (2 + |sin(1/h)| )
≤ ± 3 |h|^3
which obviously goes to 0 as h->0.
So f is differentiable at 0.