It's aweird question: at b it is not diff. because, apparently, there's a vertical line there...but then that's not a function!
At d it is clear: the function has a "shpitz" or saw-tooth point there, so fine. At g and h the function isn't defined so clearly it can't be continuous there, no matter g is a removable discontinuity...
More formally, think about how you calculate the derivative at the point .
But if doesn't exist, you can't calculate that limit.
However, if you wanted to calculate , that would exist (in this case).
What EXACTLY "isn't necessarily true"? is NOT a line segment at the origin. Since this is a visual exercise I am basing my opinion on what I see and that drawing shows like a vertical line segment at b. Of course, one could bring examples even more basic, as , which also isn't differentiable at the origin.