Now let us look at two more functions as listed on page 86 by Strang, Calculus, 1991. He lists y = 1/x^2 and y = sin(1/x).
Now this is the Uni-text and so let us use the 100 Grid rather than the 10 Grid. Remember, a grid system is where you pretend a infinity borderline as 100 for the 100 Grid and thus 0.01 is the smallest nonzero number. And of course, we use 1st quadrant only, but, for the sine function we have to realize that it is shifted to make it all in the 1st quadrant as that of y = sin(1/x) +1.
Now in Strang's Old Math, both of those functions are discontinuous, but in New Math all functions are continuous and as we shall see, since all functions are continuous, all functions have a derivative everywhere.
Now in the 100 Grid, the first few numbers are 0, then .01 then .02 then .03, then .04 etc etc.
For the function y = 1/x^2 we have:
x= 0, y = we have to wait and see what the neighboring points are
x = .01, y = 1/.00 and have to wait
x = .02, y = 1/.00 and have to wait
x = .03, y = 1/.00 and have to wait
x = .04, y = 1/.00 and have to wait
x = .1, y = 1/.01 = 100
x = .11, y = 1/.01 = 100
x = .15, y = 1/.02 = 50
Perhaps I have not explained well enough to the reader or student that in a Grid system, the only numbers that exist are those of that Grid that are finite. So for example when we multiply .15 by .15 in 100 Grid that the answer is not .0225 for that is in the 10,000 Grid, but rather .02. In Grid systems, we truncate answers to what exists in that Grid.
Now we go back to defining what division by 0 in the function y = 1/x^2 means.
We see that for x values from 0 to .09, the y value will involve division by zero and then from .1 onwards the y value drops from 100 to an asymptote along the x-axis. So that for the values of x from 0 to .09, we say the y value is 100.001 since that is an infinity number for .001 does not exist in the 100 Grid.
In New Math, we eliminate division by zero by introducing an infinity number so the graph of the function is continuous. And so division by zero is defined, but the warning is, that operation on a infinite number is undefined. And when we connect the graph points, the line segments go through the number 100 on the y axis. So for the function y = 1/x^2, we have a flat line from (0,100.001) to that of (.1, 100). We know it is slightly tilted and not flat as say y = 3 function, but we cannot operate with a .001 in 100 Grid.
Now for Strang's last discontinuous function on page 86, he talks about y = sin(1/x) and shows a blackened rectangle near zero. He claims the function fluctuates wildly as it approaches x = 0. In New Math, we have better commonsense.
Let us use the 100 Grid system for y = sin(1/x)
Now for some points:
x = .01, y = sin(100) = -.50
x = .02, y = sin(50) = -.26
x = .03, y = sin(33.33) = .94
x = .04, y = sin(25) = -.13
x = .05, y = sin(20) = .91
So in New Math, the function y = sin(1/x) is as well behaved as it approaches 0 as it is anywhere else along the x axis. I hesitate to graph the function and the student because we need all 4 quadrants and cannot do it just in the 1st quadrant only. Unless, however we shift that into the 1st quadrant as that of y = sin(1/x) +1.
But I wanted to emphasize the issue, that in New Math, as we detail the infinity borderline, it causes the inverse of the macroinfinity to define a microinfinity and leaving empty space between 0 and this first nonzero finite number. That empty space allows and demands all functions be continuous, and to connect successive points of the function graph by straight line segments that travel from one finite point to the next successive finite point. This connecting straightline segment is our derivative.