Now before I leave the topic of continuity and discontinuity of mathematics, let me provide an understanding of how awful continuity becomes in Old Math as compared to how easy continuity becomes in New Math.
In Old Math we need go no further than the two examples of the pathological functions of the y = sin(1/x) and the Weierstrass function as shown in Wikipedia that looks like a ever continuous fractal of jagged mountain peaks.
In the last post, I discussed y = sin(1/x) and when graphed in New Math, there is no aberrant behaviour of the function as it approaches x = 0. And the reason is obvious, because the points of all graphs in New Math have a spacing of at least microinfinity apart which in 10 Grid is 0.1 distance apart on the x-axis. In Old Math, they had no smallest distance apart and so they could have worlds upon worlds of crazy and fancy drawings as you get smaller and smaller. In Old Math, y = sin(1/x) is a solid blackened mass of a graph as it nears x=0, and the Weierstrass function has always another mountain peak, no matter how small you become. But in New Math, we stop and have an end of fantasy make believe of function graphs when we reach 0.1 distance in 10 Grid, or 0.01 distance in 100 Grid or 0.001 distance in 1000 Grid. The microinfinity prevents there being pathological functions of make believe.
In New Math, all functions are continuous everywhere and the microinfinity prevents functions from becoming pathological such as the Weierstrass function which was believed to be continuous everywhere but differentiable nowhere. In New Math, the derivative is the line segment that connects neighboring points together and so if a function is continuous everywhere, then it is differentiable everywhere.
Enough of the sad situation of fake math of Old Math on continuity and discontinuity. Let us instead focus on what is the worst looking function in New Math. I discussed this function in the High School text but have to repeat it here to compare with the Weierstrass function and y = sin(1/x).
I call it the Sawtooth function.
F(x) = 0 when x is even number and F(x) = 10 when x is odd number. (Here we extend the definition of even and odd to decimal numbers).
So the graph of this Sawtooth function in 10-Grid, 1st quadrant only, looks like this:
Now the function when graphed in New Math looks like this only very much steeper of triangle tops.
In fact, it would look like a row of needles lined up.
It would look like needles because the steepest angle that is not 90 degrees itself is enlisted in each Grid system as it builds a triangle whose base is from 0 to 0.1 and whose height along the x=.1 column of the cell 0 to 0.1 is 10 metric units distance.
The derivative between two points of the function are either 100 or -100, depending on direction of slope. Recall, the derivative of y = x is +1. The derivative of this function is 100 because the dy/dx is that of 10-0 for dy and 0.1-0 for dx, so you have 10/1/10 = 100. Now the derivative is a line segment in each of the cells that connects only two finite points and the entire rest of the line segment is in empty space composed of infinity points such as for example 0.01 or 0.023. Empty space is infinity points and Calculus is only about finite points being connected in a sequence. The line segment that connects is always a Euclidean straight line segment and thus the derivative is a straight line as well as the function graph. So in True Calculus there are no "curved lines".
Now this Sawtooth function is the worst it can get in New Math and its integral is simply 1/2 the area of 10 x 10 = 100 square units. So the area of the integral is 50 square units for the entire function to infinity. This is intuitive for the area under each triangle in each cell is 1/2 the area of the entire cell.
So that is the worst function in New Math, and it is continuous everywhere, differentiable everywhere and integrable everywhere.
The aspect of New Math that makes it so easy to do the Calculus, is that we are able to "zoom in and inspect" every point and its successor point in New Math. In Old Math, they had no infinity borderline and no microinfinity borderline so that when functions of y = sin(1/x) or Weierstrass functions were seen, they let their imaginations run amok and consoled themselves that the limit concept endorsed that wild and crazy viewpoint.
Now in the next post I will demonstrate how the half circle as a function in the first quadrant in the 10 Grid looks more like a half-regular-polygon than it looks like a half circle. Then the same is done for the 100 Grid, then the 1000 Grid etc etc. In one of those Grids we match the modern day computer screen that you see a circle on your computer because the straightline segments are so small that your eyes perceive the half-regular-polygon, not for its true entity as a polygon but your eyes see it as a smooth curve of a half circle.
So New Math and True Calculus are supported by pictures and diagrams of computers, for computer drawings and pictures are all based on tiny straightline segments, not what Old Math is, for it is based on both straightlines and curves and on no infinity borderline for which the limit concept is there to disguise the phoniness of Old Math.