Change in mathematics comes at a snail's pace, even when the establishment is grotesquely wrong, so I write this more for the present day teachers of Calculus, putting them to shame for the pain they inflict on young minds having to learn a pollution and not true calculus.
Alright in the 4 textbooks that I refer to of Stewart 5th ed., Strang 1st ed., Fisher & Ziebur, 2nd ed, and Ellis & Gulick 3rd ed, that they all have a chapter on discontinuity:
page 124 of 1168 pages for Stewart page 86 of 614 pages for Strang page 57 of 742 pages for Fisher & Ziebur page 88 of 978 pages for Ellis & Gulick
So that roughly 10% mark of the full pages of the texts above begins the discussion of the continuity and discontinuity of functions, and of coarse all 4 texts define continuous using the limit concept.
So in this textbook, we spare the students the agony, the misery the dismal feeling of irrelevancy that is the limit concept, to anything going on in mathematics because the people before 2011 were too ignorant or too lazy to know that once you precisely define the borderline of Finite with Infinite, that you never need a limit. The limit is a ruse, when a teacher has nothing else to say, just as a magician diverts your attention away from his hands so he can pull out a rabbit.
All functions of mathematics are continuous just from their sheer existence as functions, and if a division by zero impedes continuity, we add a pointwise patch, such as the example of y = 1/x^2.
So how does Old Math require a limit when New Math of True Calculus tosses them out the window as phony, as utterly irrelevant pollution?
Because in Old Math they never precisely defined finite versus infinite with a borderline which creates empty space between successive numbers, and thus they forever need some excuse, some fakery mechanism such as the limit concept to clean up their messes. In New Math, we have empty space between finite points which are connected in a cell by a straightline segment. That connecting insures every function is always continuous and that discontinuous functions are a breakdown of the Old Math and its phony limit concept.
The straightline that connects two points of the walls of the cell, the leftward wall and the rightward wall is actually the graph of the function itself in that small interval of x. That straightline segment is not only the graph of the function, but the derivative of the function, and a side of the 4 sided figure that composes the integral.
So, Old Math never defines finite with infinity and thus they have curves everywhere along with straightlines and they need to do Voodoo witchcraft dances every time they want to figure out if their function is continuous and what the derivative and integral are.
Whereas in New Math, and True Calculus, we simply plot the Cell for a specific x interval and connect cell walls of y with straightlines and we have the function graph and its derivative and its integral.
So, why burden and pollute the minds of young people wanting to learn True Calculus with a foolish, irrelevant limit concept, when the mathematics community is too ignorant and too lazy to change so that the students are not burdened by nonsense?
How can any college teacher of Calculus, go back into his/her classroom to teach irrelevant nonsense, because they are either too lazy or too ignorant to define the borderline of finite with infinity and thus toss the limit concept out onto the trash pile of shame.