The graph is nonstandard and misleading. It should not have a vertical segment at 0, it should have an open-circle at 0,0, and a closed circle at 0,1, showing that the lines do not and do contain the 0 point, respectively.
This makes the intuition pump a little easier. The derivative at all nonzero Xs is 0. The derivative AT ZERO, is 0 to the right (as X increases), and undefined to the left (as X decreases). There is no connection between 0 and 0 - epsilon, and therefore no slope.
You CAN use more complicated models to describe some features of it (hyperreals, or just limits), but those are modeling tools to answer different questions than the intuitive use of derivative (slope of a continuous curve). It’s probably not right to say that any of them are “true”, without some caveats.
The graph is nonstandard and misleading. It should not have a vertical segment at 0, it should have an open-circle at 0,0, and a closed circle at 0,1, showing that the lines do not and do contain the 0 point, respectively.
This makes the intuition pump a little easier. The derivative at all nonzero Xs is 0. The derivative AT ZERO, is 0 to the right (as X increases), and undefined to the left (as X decreases). There is no connection between 0 and 0 - epsilon, and therefore no slope.
You CAN use more complicated models to describe some features of it (hyperreals, or just limits), but those are modeling tools to answer different questions than the intuitive use of derivative (slope of a continuous curve). It’s probably not right to say that any of them are “true”, without some caveats.
added some open circles