Question

In the video lecture 02 between 24'~26' the function at fist didn't include x=0; then can we still take the limit of x to 0+ and 0-? Doesn't that imply that it isn't defined at x=0? or does 0+ and 0- are slightly beside 0 and not 0 yet? And if so can we use this method to differentiate discontinuous functions at some point (x0,y0)? The definition following that in the video lecture seems to suggest otherwise and then doesn't that mean that the example is false? (just had to keep watching the video and it's an example of Jump discontinuity haha)