Exercise · ZW-SMOOTH-FUNCTION-008 · ★★★ Advanced
Let \(f(x)=x^{2}\sin(1/x)\) for \(x\ne 0\) and \(f(0)=0\). Which statement is correct?
Choose one option.
Show answer
- C. Differentiable but not \(C^{1}\)
Explanation
We have \(f'(0)=0\), but \(f'(x)=2x\sin(1/x)-\cos(1/x)\) has no limit at \(0\), so \(f'\) is not continuous.
Review the concept: Smooth function
A smooth function is a function between spaces with differentiable structure that, written in local coordinates, has derivatives of all orders.
\[f\in C^{\infty}(M,N)\]
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