This is all pretty straightforward
$B(r) = \begin{cases}\dfrac{B_0}{r_0}r &r \leq r_0 \\r_0 B_0 \dfrac{1}{r} &r_0 < r \end{cases}$
So we see that $B(r)$ is linear in $r$ for $r\leq r_0$ and is proportional to $\dfrac 1 r$ for $r > r_0$
We also note that $B(r_0) = B_0$
Given above it really should be pretty clear which graph corresponds to $B(r)$ (hint: C)
Is $B(r)$ continuous at $r_0$? Well look at it. Is it? (hint: B)
Does the derivative exist at $r_0$ It should be pretty clear that D is correct. Functions with kinks in them aren't differentiable at the kink.