Exercise 5.3.4

Let $f$ be differentiable on an interval $A$ containing zero, and assume $\left(x_n\right)$ is a sequence in $A$ with $\left(x_n\right)\to 0$ and $x_n\ne 0$.

(a)

If $f\left(x_n\right)=0$ for all $n\in N$, show $f(0)=0$ and $f^{\prime }(0)=0$.

(b)

Add the assumption that $f$ is twice-differentiable at zero and show that $f^{\mathit{\prime \prime }}(0)=0$ as well.