Answer
a) For $f''(x)$ to exist, $f'(x)$ must be differentiable and continuous at $(a,b)$. Similarly, for $f'(x)$ to be differentiable and continuous, $f(x)$ must be differentiable and thereby continuous. Thus, if $f''(x)$ exists for each $x$ in $(a,b)$, then both $f$ and $f'$ must be continuous on $(a,b)$.
Work Step by Step
b) Extending this definition, for $f^{(n)}(x)$ to exist, then $f^{(n-1)}$ must exist and be continuous and differentiable. For $f^{(n-1)}$ to exist, $f^{(n-2)}$ must exist, be continuous, and be differentiable, and so on. Thus, if $f^{(n)}$ exists, than all derivatives less than $n$ must exist and be continuous and differentiable as well.
