Only I and III are correct. Remember that when I have a pointy part in a graph, then there does not exist the derivative of that part of the function. Another way to look at it, is with the absolute value function, it might help clear a little any doubt due to this problem.
This means that the derivative does not exist when x=b, but clearly the function is continuos, because if you were to draw the function with a pencil, it would not be necessary to lift the pencil in order to draw the whole function. The function is also differentiable in (a,b), this is because the point b is not included, so even though is not differentiable in b, it is in any other point close to it.
I sure hope that this helps.