Analysis | Part IA, 2003

Define what it means for a function of a real variable to be differentiable at xRx \in \mathbb{R}.

Prove that if a function is differentiable at xRx \in \mathbb{R}, then it is continuous there.

Show directly from the definition that the function

f(x)={x2sin(1/x)x00x=0f(x)= \begin{cases}x^{2} \sin (1 / x) & x \neq 0 \\ 0 & x=0\end{cases}

is differentiable at 0 with derivative 0 .

Show that the derivative f(x)f^{\prime}(x) is not continuous at 0 .

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