= Solution
The <Taylor expansion> about $a$ has coefficients given by the successive <derivatives> at $a$:
$$
f(x)=\sum_{k=0}^N\frac{f^{(k)}(a)}{k!}(x-a)^k+R_N(x).
$$
When $f$ is $N+1$ times continuously differentiable on the interval between $a$ and $x$, <Taylor's theorem> gives $R_N(x)=f^{(N+1)}(\xi)(x-a)^{N+1}/(N+1)!$ for an intermediate point $\xi$. If the remainders tend to zero, this yields the infinite <Taylor series> $f(x)=\sum_{k=0}^{\infty}f^{(k)}(a)(x-a)^k/k!$. The three <functions> below are <real analytic> near zero after taking the continuous extension in part (i).
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