I am going through MIT 1A: Differentiation course at edX. It has this graph for some stepwise-function.
PROBLEM: MIT says it is a ray when you zoom-in at (0,1), I think it looks more like a point. I can't understand how it is a ray when I zoom into it. It is just a point.
That's a weird question, but the answer is pretty straight forward: it is a ray. If you "zoom in" at the point $(0, 1)$, you're never going to zoom in far enough that the horizontal line $y = 1$ is absent from the picture. There's no tiny gap between $(0, 1)$ and the rest of the line, so you can't "zoom it away".
Remember too, of course, that although the point $(0, 1)$ is represented by a filled-in circle, it is just a point, with no width or length. Zooming in will not make the point any larger or smaller.