The so-called 'rigorous' Weierstrass epsilon-delta limit definition states that if a limit exists, then there are real numbers, epsilon and delta greater than zero, so that for every epsilon there exists a delta, such that if the distance between delta and an x-coordinate c close to the limit is less than zero, then the distance between the limit and the values of the function close to the limit(or f(c)) is always less than epsilon. What follows are some misconceptions arising from this definition:
Misconception number 1: One cannot make epsilon or delta as 'small' as one wishes. The reason is that delta must be greater than 0. This is required to ensure one can distinguish between the limit and continuity of f at x=c, and also so that Weierstrass's definition remains valid. Oddly enough, f must be defi! ned at c and f(c) must be equal to the limit to ensure continuity. Continuity at a point is not an intuitive concept. But, I hasten to add that continuity is not at issue here.
Misconception number 2: Finding an epsilon and delta proves nothing. Why? Well, one can choose an x close to c and then proceed to find epsilon and delta. Provided one keeps selecting smaller values of x closer to c, one can be certain the magnitude of either epsilon or delta for the previous larger x, will be greater than |c-x|. This analogy is so moronic, that it can easily be compared to the following: Given a rod 1 meter long. If we cut off 1/4 meters, then what remains will always be greater than 0 meters and less than or equal to 3/4 meters.
Misconception number 3: Academics find values for epsilon and delta that provide a radius of convergence around c. However, suppose that f becomes 'erratic' at some very small distance from c, how can one cont! inue to be sure the above guarantee of convergence will hold? ! Suppose that for very small distances (almost indistiguishable from c), f veers off (#) to some unknown value not equal to the limit. Can every point in the interval of convergence be checked? There are infinitely many points to check. Although the example just stated is highly unlikely, so also, are the many irrelevant examples used by academics that include piece-wise functions and other special functions to support their point of view. (#) This scenario of veering off erratically is unlikely because the concept of limit is irrevocably linked to well-behaved or smooth-continuous functions. What if a temporary discontinuity could be constructed in the form of an asymptote at some point near c? Most will respond that this is highly unlikely and probably impossible simply because an example cannot be constructed. Well, some academics believe in infinitesimal numbers yet cannot exhibit even one such number!
Misconception number 4: Use of the limit concept in the de! finition of differentiability. The limit concept has nothing to do with tangency or slope of a tangent at a point. It crept in many years after Newton because of the failure of mathematicians to find methods of computing slopes at a point. It may come as a surprise, but Newton was working with well-behaved functions. This is to say, smooth-continuous functions. Consider the following diagrams:
Figure 1 shows a well-behaved function that is both smooth and continuous whereas figure 2 illustrates a function that is continuous but not smooth. This function is superimposed on the function in figure 1. The figure 2 function can only be written as a piecewise function! . It makes no sense to talk about limits or differentiability ! in this case. Now, to see how the definition of continuity falls apart: Is the function in figure 2 continuous at x=1? Well, the left hand limit = 2 and the right hand limit = 5. Thus by definition, this function is not continuous at x=1, right? Wrong! It can be drawn without lifting one's pencil. So much for the rigorous concept of continuity defined in terms of Weierstrass's epsilon-delta limit definition.Academics today are ignorant beyond belief. Would it surprise you that many graduates (including PHds) have no sound understanding of what calculus is? Well, think about the following: Why is there such a crisis in calculus education? Do students not understand calculus because it is hard or because their teachers are ignoramuses? Why is it that such confusion abounds in so many areas of calculus (and mathematics in general)?
Related blogs:
More on limits.
Real analysis fails?
Are the real numbers well-defined?
continuity functions
No comments:
Post a Comment